-INTRODUCTION TO MATHEMATICS By C. C. T. BAKER B.Sc, Dip.Ed. A number system has always been a basic necessity for science, tech- nology and civilization, but the build-up of even the most cumbrous system was a gradual one from the days of the ancient Egyptians, Babylonians and Romans, to the Hindu -Arabic system in use by most people today. At the present time, however, the pace has altered. Rapid not gradual changes are taking place in educa- tion as in other ways of life. New approaches, ideas and techniques are appearing in mathematics which cannot be ignored. This book has been written ex- pressly to help teacher and student alike to meet the new trends and to be awake to new ways and topics which have recently appeared in some of our schools and colleges. 15/- NET Introduction to MATHEMATICS by C. C. T. BAKER, B.Sc, Dip.Ed. Lecturer in Mathematics NEWNES : LONDON © C. C.T.BAKER, 1966 First Published 19CG Printed in Great Britain by Butler tt: Tanner Ltd, Frame and London, for George Newnes Lid, Tower House, Southampton Street, London, W.O.s To my Wife Page ix CONTENTS Preface Chapter 1 NUMERATION SYSTEMS 1 Place value — the duodecimal system. 2 THE BINARY system 3 Changing from denary to binary notation — changing from binary to denary scale— operations involving binary scale numbers — exercises 1 to 6. 3 CALCULATING DEVICES 9 Computers. 4 THE BASIC LAWS OF MATHEMATICS 12 Addition — commutative law of addition— associa- tive law of addition — multiplication — commutative law of multiplication — associative law of multiplica- tion — distributive law of multiplication — zero — unity — zero and unity compared. 5 sets 1? Set notation - -membership of a set — kinds of sets — one-to-one matching — Venn diagrams — operations with sets. 6 THE ALGEBRA OF SETS 24 Commutative law — associative law — distributive law — De Morgan's laws — law of absorption — laws of Boolean algebra — summary of the laws of Boolean algebra — exercise 7. 7 THE GEOMETRY OF SETS 33 Union and intersection of lines. 8. grouts 37 Group structure of the integers — rings — fields. 9 TRANSFORMATIONS OR MOTION GEOMETRY 39 Reflection — rotation — rotational symmetry— trans- lation — shearing — s irailari ty . 10 TOPOLOGY 45 Classification of topological figures — Euler's formula — Euler's networks — Moebiua strip — Klein bottle — map colour problem — three-to-three connections — exercise 8. 11 THE ALGEBRA OF CIRCUITS 56 Tables — two-way switches — Boolean algebra and switching circuits— exercise 9. 12 DETERMINANTS IS") Rule of Sarrua — solution of equations — properties of determinants — exercise 10. 13 MATRICES 72 Multiplication — addition — subtraction — multiplying a matrix by a factor — equal matrices — unit matrix — zero matrix — diagonal matrix — inverse matrix • finding the inverse of a 2 x 2 matrix — simultaneous equations using matrices— applications of matrices — rotation through 180° — exercises 11 and 12. 14 vectors 86 Kinds of vector — addition of vectors — theorems on vectors — multiplication of a vector by a scalar — subtraction of vectors — scalar product — theorems on scalar products — vector products — exercise 13. 1 5 INEQUALITIES 96 Pictorial representation of inequalities — intersecting graphs — exercise 14. 16 FINITE ARITHMETIC 101 Residue classes modulo 3 — summary — residue classes modulo 6 — residue classes modulo S — exorcise 15. 17 SYMMETRY GROUPS 106 Symmetry operations — identity element and inverse element — group of symmetries of the rectangle — symmetries of the equilateral trianglo — isomorphic groups — exerciso 16. 18 SENTENCE LOGIC 113 Propositions — compound propositions — negation of a proposition— implication — statements represented symbolically — truth tables — illogical reasoning — ■ equivalence — Boolean algebra, sets, and logic — exercise 17. 19 POINTS, RELATIONS AND FUNCTIONS 122 Ordered pairs — relations — lines and regions- -func- tions — correspondence— inverae correspondence — mapping — exercise 18. 20 STATISTICS 127 Pictorial representation of data — moan value — mode median— relation between mean, median and mode — percentile rank — scattering, range, deviation — exercise 19. 2 1 PROBABILITY 1 35 Addition law — multiplication law — probability tree — exercise 20. List oJSymbola 130 Definitions 140 viii PREFACE Long ago, changes took placo slowly and gradually. There was little difference in the way of life, even from one century to another. At the present time, changes are taking place with increasing rapidity, and there is often a noticeable difference in some aspects of life from year to year. This applies equally to education. Schools and the subjects taught in them have changed greatly during the lifetime of living people. Today, the whole system of education in this country and others is being altered. New approaches, new ideas and new techniques are appear- ing in mathematics. Teacher and student aliko should be awake to these new ways so that they can be up-to-date hi their knowledge. Many of these changes have come about through change in outlook duo to tho modern demands of industry and science. This hook doals with some of the topics which have recently appeared m some schools and colleges. It is hoped that it will be a useful introduction to the new trends. C.C.T.B. CHAPTEE ONE NUMERATION SYSTEMS The development of a number s ysfcem is, and always has been, of fundamental necessity for the purpose of science, technology, and civilization. Even primitive people needed a method of counting. Then* method was, what is now called, one-to-one correspondence. Suppose goods exchanged hands, and imagino the goods to be cows. Each cow would be represented by one notch on a stick. Mathematicians now call this matching operation a 'mapping'. In this particular case, each member of the 'set* of cows is mapped on to each member of the 'set' of notches. This operation also gives us the idea of a 'cardinal* number, because the symbol 8, in 8 cows, is one of the set of cardinal numbers. The 8 is distinct from the 8 in 'the 8th cow*, because the latter is what is called an 'ordinal' number. It took very many centuries to build up even the moat cumbrous number system. There have been, in the history of man, many different kinds of number systems. To mention a few, there were those of the Egyptians, Chinese, Babylonians, and Romans. In some systems, each number is represented by an entirely different symbol, each symbol being called a numeral. This makes for a very large number of Bymbols. The Roman system was, in many ways, very primitive, because the numbers 1, 2, 3, 4, were represented by strokes, |, | |, | | j, | | | |. The Chinese used horizontal strokes, — >■>,■*,■• Today, wo use the Hindu- Arabic system. This was invented in India, and it was brought to Europe by the Arabs. Nowadays, most people accept it as boing the only possible system of numera- tion. In fact, it may not be the best, and the time may come when a different and better system will be found. Even now, for some purposes, other systems are already being used. For example, the binary system is employed for computers. place VALUE The invention of a place value system was the most revolutionary of all, as it reduced considerably the number of symbols needed. In the denary system, which is the one now in common use, the symbols aro; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is 1 called the base ten system. The zero, 0, was introduced by the Hindus of northern India. Piimilive man had no need of a symbol for zero, because he used numerals as recording sym- bols, nob as working symbols. Zero is essential in any place value system. Now to explain the meaning of 'place valuo'. Consider the denary number 333. Each 3 has a different value. The 3 on tho left stands for 300, the 3 in the middle stands for 30, and the 3 on the right stands for 3 units. In fact, the number 333 could be written as: 333 = 3 x 10' + 3 x 10 l + 3 x 10°. Similarly, a number such as 5387 could bo written as : 6387 = 5 x 10 s + 3 x 10 1 -f 8 x 10 1 + 7 x 10 B . It will be seen that, as a number moves towards the left its value increases tenfold, and the indices belonging to the base 10 increase by one. The denary system is sometimos called the decimal system, and decimal fractions can be treated like the integers, so the fraction 0'8349 could be written as : 0-8349 = 8 x 10-* + 3 x 1Q"» + 4 x 10~ 3 + 9 x 10-«. It could be mentioned here, that the 0, in 8349, may be regarded as a place holder. Its use will be seen to be very necessary in a number such as 8309. This Hindu-Arabic numeration system has vast superiority over the Roman system, especially in, even simple, multiplication and division. THE DUODECIMAL SYSTEM This system is based on the number 12, mainly because there are 12 inches in ono foot. It is often used by carpenters and by builders. There are, however, other uses for a base 12, as in (a) 12 x 12 make a gross, (b) 12 eggs make one dozen, (o) 30 x 12 degrees make one revolution, (d) 5 x 12 minutes make ono hour. If a duodecimal system were universally adopted, it would be necessary to introduce two new symbols, ono for ton, say I, and one for eleven, say e. The twelve symbols would then bo : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, t, e. CHAPTER TWO THE BINARY SYSTEM This system was first advocated in the seventeenth century, by the German mathematician Leibnitz, because it uses only two symbols, and 1. Strangely enough, some vory primitive people used it long before the seventeenth century. Like so many topics in mathematics, binary numbers dropped out of interest for a long time, but the appearance of the elec- tronic computer has brought new uses for them. This is the reason. In an electronic circuit, there are two states to consider. Either a current is flowing, or it is not flowing. Thus, the num- ber 1 can represent a circuit with the switch 'on', and the number can represent a circuit with the switch 'off'. The principle of placo value is used with binary numbers in the same manner as it is used with denary numbers. Using the denary system, the place values are such as 10* -|- 10 1 + 10* + 10* + 10°, etc. Using the binary system, the placo values are such as 2* + 2 3 + 2' + 2 1 + 2», etc. TABEE COKNEOTTSfO DENABY AND BINAHY NUMBERS Denary Index form Binary form foj'tn 1 1 X 2° 1 2 1 X 2 1 + x 2" 10 3 1 x 2 1 + 1 X 2" 11 4 1 X 2* + X 2* + X 2° 100 6 1 X 2» + X 2> + 1 X 2° 101 6 1 X 2* + 1 X 2 1 + X 2° 110 7 1 X 2* + 1 X 2' + 1 X 2" 111 8 1 X 2' + X 2" + X 2 1 + X 2" 1000 9 1 X 2» + X 2* + X 2 1 + 1 X 2» 1001 When a number is written down, only the coefficients are stated, as in the comparison table on pago 4. PLACE VALUE COMPARISON" TABLE Denary Binary 100 10 1 1 2 3 64 32 1C 8 4 2 1 1 ll 1 1 5 1 1 1 1 2 9 1 1 1 1 3 7 1 1 1 5 1 1 1 1 1 1 2 3 1 1 1 1 1 1 TO CHANGE FROM DENARY TO BINARY NOTATION Metltod. Find the largest multiple of 2 in the number, subtract thia multiple of 2, repeating the process with the remainder. Example. Change 1 3 in the denary system to the binary system. (i) The highest multiple of 2 in 13 is 8, which is 1 X 2*. Subtracting this multiple of 2; 13 — 8 = 6. (ii) The highest multiple of 2 in 5 is 4, which 1b 1 x 2*. Subtracting this multiple of 2: 5 — 4 = 1. (iii) Therefore, 13i = 1101 2 . 13, means 13 in the denary system. So 13 lD is the same as 110I 2 , which represents 1101 tn the binary system. Further Examples: 1. "in = 1 X 2' + x24 1 x 2° = 101*. ■1. ",. = 1 X 2* + x 2* + + 1 x 2* + x 2° X 2 1 _. 100Q1 2 . 3. 38 10 = 1 X 2 s + x 2* + x 2 a + 1 X 2* + 1 x 2 1 +0 X 2° ■ ioouOj KXEBCISE 1 Change the following denary numbers to binary numbers: 1. 7 2. 9 3. 12 4. 20 5. 38 0. 50 7. 53 8. 64 9. 121 10. 256. ANSWERS TO EXERCISE 1 1. Ill 2. 1001 3. 1100 5. 100110 6. 110010 7. 110101 9. 1111001 10. 100000000. 4. 10100 8. 1000000 TO CHANGE FROM BINARY TO DENARY SCALE Method, (i) Multiply the left-hand digit by 2. (ii) Add on the next digit to tho right, (iii) Multiply the result by 2. (iv) Add on tho next digit to the right. And repeat this process. Example. Change the binary number 111 to a denary number. Write the number down. 111. (i) Multiply tho loft -hand digit by 2 and get 2. (ii) Add the 1 to the right: 2 + 1 = 3. (iii) Multiply the result by 2 : 3 x 2 = 6. (iv) Add on the next 1 to the right: 6 + 1 = 7. Therefore, 111. - 7 W . Further Examples: 1. 11 2 = 1 x 2 + 1 = 3 10 .'. 11, = 3 W 2. HOlj = 1x2 + 1= 3 3x2+0= 6 6 x 2 + 1 = 13 .". 1101, = 13 10 . EXERCISE 2 Change the following binary numbers to denary numbers. 4. 10111 1. 101 5. 11011 9. 101 2. 6. 10. 110 11111 11-01 1101 110111 8. 111101 ANSWERS TO EXERCISE 2 1. 5 2. 6 3. 13 6. 31 7. 55 8. 61 4. 23 9. 2£ 5. 27 10. 3J. OPERATIONS INVOLVING BINARY SCALE NUMBERS (1) ADDITION The main points to remember are : (a) 1 + = 1 (b) 1 + 1 - 10. Example 1. 11 10 101 .*. 11 + 10 = 101 Example 2, 1011 101 10000 .'. 1011 + 101 = 10000. 5 EXERCISE 3 Add the following binary numbers. 1.11 + 11 2.100 + 11 3.110 + 11 4.111+11 -,. 1001 +1 6. 1101 + 10 7. 1011 + 10 8. 110110 + 101 9. 1111 + 111 10. 11101 I- 11. ANSWERS TO EXERCISE 3 1. 110 2. Ill 8. 1001 6. 1111 7. 1101 8. 111011 (2) SOBTBAOTION The main points to remember are 4. 1010 9. 101 10 6. 1010 10. 100000. (a) 1 Example 1. Example 2. Example 3. 1=0 (b) 1 - = 1 (c) 10 - 1 = 1. 101 11 10 110 11 11 11100 101 10111 exercise 4 Subtract the following biliary numbers. 1. Ill - 100 2. HI - U 4. 100 - 11 5. 1101 - 111 7. 1000 - 111 8. 10001 - 111 10. 10000 - 1001. 3. 110 - 101 fi. 1010 - 101 9. 1000 - 101 ANSWERS TO EXER01SE 4 1. 11 2. 100 3. 1 4. 1 e. no 6. 101 7. 1 8. 1010 9. 11 10. 111. (3) MULTIPLICATION The main points to remember are: (a) 1 x = (b) x 1 = (e) 1 x 1 = 1 (d) x = 0. The process is tho same as that applied to ordinary denary numbers. Also, to multiply by 10, simply add a 0. Thus, 101 x 10 = 1010. Also 101101 x 100 = 101101. Example 1. 101 x 11. Example 2. Ill x 101 101 11 Tolo 101 mi in 101 11100 111 100011 EXERCISE 5 Multiply the following binary numbers. 1. 1011 x 101 2. 1011 x 110 4. 1011 x 111 5. 1111 x 1100 7. 11001 x 1101 8. 11 x 11 10. 1010 x 11-01 1001 1110 1101 x 111 x 1J01 x 1 01 ANSWERS TO EXERCISE 5 1. 110111 2. 1000010 3. 111111 4. ioo no i G. 10110100 6. 1111110 7. 101000101 8. 100- 1 9. 1000001 10. 100000-10. (4) DIVISION The four operations on binary numbers are all simpler, but longer than on denary numbers. Division proceeds the same in both scales. To divide by 10, the decimal point is moved one place to the left, and to divide by 100, the decimal point is. moved two places to the left. Example 1. 1100 4- 10 = 110 1011 11010 =• 101 and remainder U Example 2. 1011 + 1000 = Example 3. 101)11010(101 101 no 101 i ITM— B EXERCISE 6 Divide the following binary numbers. 1. 10100 ■* 101 2. 1001 ^ 100 4. 10010 ~ 110 5. 11000 - 110 7. 1111 4- 101 S. 11110 * 101 10. 11011 * 1001. 3. 10101 -^ 11 6. 10101 + 11 9- 10110 4- 1011 AM8WHB3 TO EXERCISE 6 1. 100 6. Ill 2. 1001 7. 11 3. Ill 8. 110 4. 11 9. 10 6. 100 10. 11. CHAPTER THREE CALCULATING DEVICES Calculating devices have a very long history, dating from the time of primitive man. The first calculating machino was probably the human hand. It is quite possible that tbo denary system owes its existence to the fact that the human being has ten fingers. Most, if not all, children begin to count with the nid of the fingers. The abacus, an arrangement of wires holding rows of coloured beads, has been in existence for centuries. It is still popular with children. The early Chinese used the abacus, and other kinds of counting boards. More recently, John Xapier invented a dovico called Napier's rods, or Napier's bones. Napier invented one of the first exponential calculators, the natural logarithm, the base being e. John Briggs developed common logarithms, to base 10. The slide rule, so much used by engineers, is an exponential calculator, and its use depends upon the laws of indices. (1) Since a m X o B = o m +", it will be seen that, if the log- arithms of numbers are added, the logarithm of the product is obtained. (2) Since a m 4- a n = a m - n , it will be seen that if the log- arithms of numbers are subtracted, the logarithm of the quo- tient is i itii ained. The slide rule makes use of this property of logarithms. COMPUTERS As civilization has advanced new apparatus and new devices, new materials and new processes, have been invented. Now, even space outside our planet is being explored. The means for doing these things have depended more and more on compli- cated and lengthy calculations. Some of the calculations in- volved would have required thousands of mathematicians working for many years. Fortunately, calculations can now be performed extremely rapidly with a computer, for it is capable °f performing highly complex operations at the speed of light . Although a computer is a very complicated piece of equipment, it can do only three things: (i) add two numbers, (ii) subtract one number from another number, and (iii) compare the size °f one number with that of another. Computers are of two types, the analogue computer, and the digital computer, 1. The analogue computer makes use of variations in electric current and voltage to represent numerical values. Very often, in tho physical world, quantities are connected by the same kind of formula, even though the quantities bear no connec- tion with each other. Thus, Newton's Law, p = mf, connecting force, mass, and acceleration, is the same kind of formula as Ohm's Law, V = Jit, connecting voltage, current, and resis- tance. This means that an electric circuit, by varying current and voltage, can measure variations in an 'analogous' system. Analogue computers, however, are less accurate than digital computers. 2. The digital computer performs calculations as one would do problems in arithmetic, using an abacus. Before a computer can deal with a problem the data has to be 'programmed' by a mathematician known as a programmer. Ho expresses the mathematical steps involved into a code or language suitable for the machino to deal with. This may be dono on punched cards, on paper tape, or on magnetic tape. Although punched cards can be bulky, they are useful for keeping a record of data constantly being used. Paper tape is fragile but it is less bulky, it can be read quickly, but it cannot be altered if a mistake has been made. Magnetic tape usually consists of a ferric oxide fixed between two layers of a plastic. It is about 05 in. wide, and contains about 9 channels. The code consists of a series of magnetized spots, corresponding to the holes on a paper tape. The programmer breaks down the problem into a succession of additions and subtractions; multiplication being carried out by repeated addition. Some computers can add, or subtract, hundreds of thousands of times each second. The main parts of a computer are the arithmetic unit in which the actual cal- culating takes place, a storage or memory unit where previous results are collected, and a control unit which ensures that the calculations are performed in the correct order. The parta of a computer are extremely complicated, and resemble thousands of television circuits. Figure 1 shows a very simple arrangement of tho units. The problem may be mathematical, commercial, industrial, or even human, such as managerial. Whatever it is, it is first codified by the programmer. These codes are split up into small steps and are fed into the machine in the form of electrical impulses, similar to the dialling of a telephone. The impulses are created 10 by t lie input unit: they are generated by the programme which is constructed like a typewriter, and are recorded on tho tape which may be of paper or of metal. The control unit arranges tho route which tho impulses will take, and it automatically prepares the parts of tho machine which will bo needed, and cuts out those which will not be re- quired. The memory unit stores information which is being continually used. THE PROGRAMMER A MATHEMATICIAN INPUT UNIT CONTROL UNIT STORAGE OR MEMORY UNIT ARITHMETIC 1 UNIT D OUTPUT UNIT SOLUTION TO PROBLEM -"! Fig. 1 Tho solution to the problem eventually reach oa the output unit in the form of electrical impulses which aro changed into holes in paper tape, into magnetized spots on steel tape, or into hole3 in punched cards. It then needs a programmer to decode the result to give the solution to the problem. 11 CHAPTER FOUR THE BASIC LAWS OF MATHEMATICS ADDITION Wo already possess the set of symbols J, 2, 3, 4, 5, 6, 7, 8, 9, which are called numbers. If two members of this set are chosen then, associated with this pair, there is smother member called the sum. Let tho numbers 3 and 5 be chosen, then their sum is 8. The three and the five are chosen in a certain order, and we then have the ordered pair (3, 5). Addition is a mapping, since it maps a set of ordered pairs into a sot of single numbers. The mapping characteristic can be illustrated symbolically as (3, 5) "£• 8. The sign + indicates that tho process of addition mapping is performed. A mapping which assigns to each ordered pair of objects in a set another object in the same sot is called a binary opera- tion. Since the sum is in the same set as the element, tho operation is said to be closed under addition. Since there are many ordered pairs which can be mapped on to the same image, for example, (2, 6) ~+ 8, addition is said to be a many-to-ono mapping. COMMUTATIVE LAW OS- ADDITION The ordered pairs (3, 5) and (5, 3) have both the same image. If the members of an ordered pair change places, they are said to commute. Since the sum is the same, addition of the natural numbers is a commutative operation. Symbolically, the com- mutative law of addition is expressed as a + b = 6 + a . This statement may appear trivial, but that is because we are accustomed to the fact that, say, 6+4 = 4 + 6. However, familiarity can be misleading, and it is important to point out that not all tho elementary rules of arithmetic are commuta- tive For example, division is not commutative because £ is not the same as *. It is not possible to commute the 8 and the 4. THE ASSOCIATIVE LAW OF ADDITION Consider the sum of the three numbers 4, 6, 8. It may be found hi two ways, (i) Add the 4 and the 6 to make 10, and then 12 add the 8 to this total to make 18, which is the sum. (ii) Add the 6 and the 8, to make 14, and then add tho 4 to this total to make 18, again. This means that it is possible to associate the numbers in any order without affecting their sum. This is called the associative law for addition. Expressed sym- bolically, it Btatcs that : o + 6 + c = (a + 6)+c=o + (6+ c). Addition is only one of the binary operations which pos- sesses this associative property, but not all binary operations possess it. For example, tho process of finding tho arithmetic moan, or average, of numbers does not obey the associative tew. Let the symbol © be used to stand for 'find the average'. Then (i) 6 © 8 = 7 and (ii) 4 © 8 = 6. However, (a) (6 © 8) © 4 = 7 © 4 = 51, and, (6) 6 © (8 © 4) = 6 © 6 = 6. Therefore, (6 ® 8) © 4 is not the same as 6 © (8 © 4). MULTIPLICATION Young children often perform the operation of multiplica- tion by drawing a rectangular array of dots, the number of rows corresponds to one of the numbers to bo multiplied and tho number of columns corresponds to tho other number. For simplicity, suppose it is required to multiply 3 by 4. Then, the arrangement of dots would be as in Fig. 2. Fio. 2 The total number of dots gives the product of tho two numbers. In general, to multiply the number a by tho number 6, it is necessary to find the cardinal number of a set consisting of a rows each containing b objects. A product is represented by a dot, or by the sign x , so the product of a and b is a. b or a x b. This process of finding a product is a mapping of natural numbers into a natural number, and it is expressed symboli- cally as {a, b) ~J^ a.b. As an example, (3, 4) "J* 1 12. Since this mapping is defined for every ordered pair of natural numbers, and tho image is always a natural number, the operation of multiplication is a binary operation. 13 COMMUTATIVE LAW OF jnjLTIPLJCATION It is well known that numbers may be multiplied in any order, because 3x4 = 4x3, and 5 x 6 = 6 x 5, so that, generally, a x b = b x a, ora.b =6.o. This is the commuta- tive law of multiplication. THE ASSOCIATIVE LAW OF MULTIPLICATION Extending the process of multiplication to three numbers, it is well known that, (i) 3 x (4 x 5) = 3 x 20 => CO = (3 x 4) x 5. Also, (ii) 4 x (5 X 7) = 4 x 35 = 140 = (4 x 5) x 7, and so en. In general a x b x a = a x (6 x c) = (a x b) x c. This is the associative law of multiplication. THE DISTRIBUTIVE LAW OF MULTIPLICATION This law forms a link between the multiplication and addi- tion of natural numbers, and expresses tho fact that multipli- cation is distributive with respect to addition. Tho law can bo illustrated by the drawing of a rectangular array of dots. Consider an array consisting of 3 rows, each row containing 11 dots (Fig. 3). Wta. 8 The total number of dots is 3 x 11 = 33. This single array may bo split up into two arrays (Fig. 4), Fio. 4 One array has 3 rows, each containing 5 dots and the other has 3 rows each containing 6 dots. Tho total number of dots is then 3x5+3 x 6. Therefore 3x5+3x6 = 3x11, or, 3 x (5 + 6) ■= 3 x 5 + 3 x 6. In general, if a, 6, and c are natural numbers, ox(6+c)=ox6+oxc. This is the distributive law, and it shows that the multiplier can be distributed among tho individual terms of tho addition. 14 In this law, however, multiplication and addition cannot change places, because addition is not distributive with respect to multiplication. This means that : 3 + (5 x 6) is not the same as 3+5x3 + 6. The above laws have been explained by using natural num- bers as examples. However, tho Jaws would hold for any other abstract set of symbols, because cardinal numbors havo boon used which are completely detached from objects of any kind. Tin's shows that it is possiblo to oonstruct a very large variety of number systems, and these may all bo defined thus: A number system is any collection of objects on which binary operations called addition and multiplication are denned such that addition is commutative and associative, multiplication is also commutative and associative, and multiplication is distributive with respect to addition. ZERO This is a very important concept. Early man had no use for zero, but with the invention of place value in numeration systems, it became necessary to have a 'place holder'. Zero was first thought of by the Hindus, and, later, the Arabs repre- sented it by the symbol 0. When zero became a part of our present number system, it meant that zero had to behave in a manner which was consistent with that number system. Zero has the following properties: 1. Zero plus any number gives the same number again, i.e., + x = x. 2. Zoro times any number gives zero. i.e„ x x = 0. Although zoro has the above properties for our number system, they may not be true for other number systems. From property 1, zero may be called the 'identity element* for addition because, sb + = 0+*=*, for all x (written, y x). Unity It has been shown that if the zero element is added to another element, the other element remains unchanged. There is another natural number which bears a similar relationship to the process of multiplication. This is the number 1, tho unity element. It obeys tho rule : 1 x x = x, for all x, 15 Thus, multiplying a number by I leaves that number un- changed. Elements, in whatever number system, which behave like this are called unity elements. The identity law for multiplication is, x.l = 1.x = w, for all x (written y x) ZERO AND UNITY COMFA11ED Both of those have the same property. One has it in relation to one operation, and one to another operation. They are both examples of identity elements, because, if, in any system possessing a binary operation symbolized by ©, there is an element e which has the property e © as = x, for all a; in the system, then e is called an identity element, e comes from the German word einfieit, which means unity. 10 CHATTER FIVE SETS There is nothing difficult about the concept of aset. It is simply a collection or a group of things, people or numbers. The term could be applied to a family, or a team, or a pack of cards, a flock, a school, a nation, the consonants, and so on. Those items which belong to a set are called its members, or its elements. It is not necessary for the things ha a set to have any connection with each other, hut it must be possible to dis- tinguish members from non-members. The theory of sets was probably first developed by Cantor, the Gorman mathematician. It did not receive much support at first, but recently its use has been extended to many branches of mathematics, industry, commerce, and logic. SET NOTATION As the general study of set theory has only recently begun to be studied, there is as yet no standardized symbolism, and different authors often use different notation. However, that adopted here is the most commonly used. A set is symbolized by a pair of braces or curly brackets { }, or by parentheses ( ), or square brackets [ ]. Members of tho set are either placed inside or they are described inside. Thus, the set of odd numbers between nought and ten would be indicated as {1,3, 5, 7, 9}. The set of multiples of 5 from to 31 may be indicated as {5, 10, 15, 20, 25, 30}. A set may be specified by stating some property of its members, such as {c, c is a circle). Sometimes a single capital letter is used to denote a set. A could denote all the vowels, so A ■= {a, 6, i, o, u). E could mean a set of even numbers, E = {2, 4, 6, 8, 10}. The method of describing a set should show clearly which items belong to the set and which do not. MEMBERSHIP OF A SET Tho symbol used for 'is a member of * is e. e is the Greek e, and it is used because e occurs in 'is an element of, element and member being synonymous. If the set N is the set of natural numbers 1 to 9, so that N = {1, 2, 3, 4, 6, 6, 7, 8, 9}, then it may be said that 6 e N. It may also be said that 7 e N. 17 The symbol used for 'is not a member of is f. Therefore, it may be said that O^iV. Similarly, if A stands for the set {a, e, i, q, w}, then b £ A. KINDS OF SET 1. A set is countable, or denumerdble, if all its members can be arranged in so mo order, one member being first, one second, and so on. 2. A finite set has a definite number of members. It is not neces- sary for the actual number to be known. The number of grains of sand on the earth, although immense in number, is finite. AH finite sets are countable, at least in theory. 3. An infinite set has an infinite number of members. The set of whole numbers is an infinite sot. It does not follow that all infinite sets are uncountable, or non-denumerable. The set of points on a line, or the set of prime numbers, or the set of odd numbers, are examples of infinite sets. 4. The empty xet, or the null set, has no members. It is denoted by { }, or by the symbol 0. It is similar to zero in the common numeration system. Men 10 ft tall, cats with four tails, triangles with four sides, are empty sets. 6. The universal set, or the universe, is denoted by the capital letter U. This set contains all members, although it has to be described carefully and specifically. The following could be universal sets: all the pupils in a class, all Englishmen, all the books in a library. C. Subsets are sets within sets. A subset is part of a set. The set of pupils with brown eyes would be a subset of the set of pupils in a school. The set of pupils who studied Latin would also be a subset of all the children in a school. The set of pupils with brown eyes could be a subset of those pupils who study Latin. Also, those who study Latin could be a subset of those with brown eyes. Tho symbols for a subset are z> and c , The symbol ^ means 'is a subset of, and the symbol c means 'has as one of its subsets'. Let the set A = {1,2, 3, 4, 5, 6}, and let the set B = {1, 2, 3, 4}. Then B => A, and A <- B. If T is the set of all triangles (an infinite set), and if E is tho set of all equilateral triangles (also an infinite set), thou E => T, and T <= E. 7. Equal sets. Two sets are compared by comparing the 18 members of one set with the members of the other. The set {a, b, c, d) and tho set {p. q, r, s} are different becauso their members are different. However, the set {a, b,e,d) and the set {c, d,a,b) are equal sets because they have the same members. It does not matter if the members are not in the same order. Any two sets with the same members are equal. If the members of a set arc listed it is easy to compare them. Comparison is not so easy if the members are merely described. 0:>rE-TO-0NE MATCHING, OR ONE-TO-ONE CORRESPONDENCE Two sets can bo compared by matching the members of one set with the members of the other set. Let the set X = {a, b, c, d}, and let the set Y ™ {p, q, r, s). The members of set X are not the same as the members of set Y, although both sets have the same number of members. To each member of set X there corresponds a member of set Y. a i> c d MM Fio. P ? r j Figure 5 shows that there is a one-to-one correspondence. When this occurs, the sets are said to be equivalent. VENN DIAGRAMS or SET DIAGRAMS These are drawings which wore invented by the mathema- tician John Venn in order to help one to understand the algebra of sets. The universal set is usually represented by a rectangle, and it is called the sot U. A subset is usually represented by a circle, and it is denoted by a capital letter other than U. Any number of subsets may be inserted in the rectangle and they may, or may not, overlap. There are the following possibilities : 1. Disjoint Sets. Figure shows the universal set U containing tho subsets A and B which do not overlap. No member of set A is a member of set B, and vice versa. Fio. 6 Example. U is the set of nil human beings. B is the set of all boys. G is the set of all girls. This is illustrated in Fig. 7. Fio. 7 2. Intersecting Sets. U is the sot of all the pupils in a class who study French, Latin and Gorman. F is the set of pupils who study French and German. L is the set of pupils who study Latin and German. The shaded portion represents the pupils who study French, Latin and German {Fig. 8). Fio. 8 3. Subset of a subset. Example. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. A = {1, 2, 3, 4, 5, 6, 7, 8}. B = {6, 7, 8}. A is a subset of U, and B is a subset of the subset A. This is illustrated in Fig. 9. Fig. 9 4. The Complement of a Set. Let a set bo represented by A . The complement of the set A is 'not A'. It contains those members not in A. It is denoted by A' (Fig. 10). 20 Fio. 10 OPERATIONS WITH SETS There are two main operations which can be performed with sets. These are: (a) The union of sets, and denoted by the symbol U (read cup). (b) The intersection of sets, and denoted by the symbol n (read cap). 1. The Union Operation. The union of two sets is another set formed by taking as its elements those elements that are in one at the other of the two sets being united. In a number system, the union operation would be the 'addition* operation. Example 1. Let set A = {1, 3, 5, 7} and let set B = {2, 4, 6, 8}. Then A\JB = {1,2, 3, 4, 5, 6, 7, 8). This is illustrated in Fig. 1 1 . Fio. U A<jB Example 2. Let set P = {a, b, c, d, e} and let set Q => {c, d, e,f}. Then P U Q - {«. &> c, d, e,f). Tins is illustrated in Fig. 12. Fig. 12 Although the letters o, d, and e occur in both seta they are not written twice. P U Q is the set which includes all the members of the sets on which tho operation is performed. 2, The Intersection Operation. The intersection of two sets is another sot formed by taking a3 its elements all those elements which are in both subsets beuig intersected. In a number system, the intersection oporation would be the 'multiplication' operation. Example 1. Let set X = {a, b, c, d, e} and let set Y = {c, d, e,/}. Then X n Y = {c, d, e}. This is illustrated in Fig. 13 whero the shaded portion repre- sents X O Y, Fig. 13 The intersection process is performed when the H.C.F. of several numbers is being found in arithmetic. Example 2, Let set .4 => {2, 3, 5} and let set B = (3, 5, 7}. Then 4ni)={3, 5}. CD Fro. 14 AnS To find the H.C.F. of 30 and 105, arrange the numbers in their factors : 30=2x3x6, 105 = 3 x 5 x 7, Therefore H.C.F. of 30 and 105 - 3 x 5 = 15. The complement of a Set. Let U be the universal set, Fig. 15, and let A he & subset. Then the shaded portion represents the complement of A, and it is denoted by tho symbol A'. 22 This means that the set A together with the complement of A {i.e., A') mako up the universal set. The complement of the set A could be described as 'not A'. Fio. 15 Example. Let U represent the set of all the letters of the alphabet. Let A represent the set of all vowels. Then A' will represent the sot of all tho consonants. Examples on Sets. Consider the following sets : 1 = {x, y, z), a = {y, a}, 6 = {z, a;}, « = {*. y)> d = {X}, e = {y}, / - {*>■ = { }. Then: 1. a + b = a U b = {x, y, s) = 1 2. c 4- e = c U e - {x, y} = c 3. /+1-/U1- {x,y,e} = 1 4. d + d - d U d «= {x} = d 5. a X b = a n b — {:} -f G. c x c = c Pi c = {x, y) = c 7. c x / = c n / = { } = 8. 1 x a = 1 n a = {y, z} = a 9. a' = {x} -d 10. 6 + e' = 6Ue' = {x, z} = 6. 23 CHAPTER SIX THE ALGEBRA OF SETS L the commutative law This is tho law concerning Iho order in which quantities are taken, (a) For Addition, or Union (i) In arithmetic, 3 — 4 = 4 4- 3 (ii) In Aigobra, a -f b = b + a (iii) In set theory, A\J B = B U A. (iii) is illustrated thus: Let set A = {a, b, c, d } (Fig, 16), and let set B = \c,d,e,f). Then A U B = {a, b, c, d, e,f}, and B U A = {a, b, c, d, e,/}. Therefore, A U B = B\J A. Therefore, the order in which seta are added is immaterial. Fia. 16 (b) For Mtdtiplication, or Intersection (i) In arithmetic, 3x4 = 4x3 (ii) In algebra, a x b = 6 x a (iii) In set theory, A n B = B C\ A. (iii) is illustrated thus: Let set A = {a, &, c, d} (Fig. 17), and let set B = {c, d, e,/}. Then inB = {c, d}, and B C\ A = {c, <*}. Therefore, A H B - B Pi A. Therefore, the order in which seta are multiplied is immaterial. Fio, 17 2. THE ASSOCIATIVE LAW This is the law concerning the way in which quantities are grouped. (a) For Addition, or Union (i) In arithmetic, (3 + 4) + 5 = 3 + (4 + 5) (ii) In algebra, (a + b) + c = a + (b + c) (iii) In Bet theory, {AUB)\JC = A V (B U C). (iii) may be illustrated thus: Consider the following sets (see Fig. 18). A = {a, b, c, d}, B = {&, c, d, e,/}, C = {c, d, e, g, h). A\J B = {a, 6, c, d, 6, /} (AUi*)UC= {a, b, c, d, e,/, g, h} . . (i) £UC = {b,c,d,e,j,g,h} AU(BUC) - {a, 6, c, d, e,f, g, h} . . (ii) Comparing (i) and (ii), (iUB)UC -iU(BUC), Similarly, it may be shown that (A\J B)\J C = C \J (AKJ B). Therefore, the order in which sets are added does not affect the result. Fig. 18 (b) For AluUiplication, or Intersection (i) In arithmetic, (3 x 4) x 5 = 3 x (4 x 5) (ii) In algebra, (a x 6) x c = a x (6 x c) (iii) In set theory, {A A B) H C = A f\ (B C\ C). (iii) is illustrated as follows: Consider tho following seta (see Fig, 19). A = {a, b, c, d}, B = {b,c,d,e,f}, - {c, d, e, g, h). AC\B = {&, c, d} UnB)r\C = {b, c, d] n {c, d, e, g, h} « {c, d} BnC = {c,d, e} Ar\[BC\C) = {a, 6, e, d} n {c, d, c} = {c, d}. Therefore, (4 n B) O C = ,4 Pi (B H C). 25 Fio. 19 3. THE DISTRIBUTIVE LAW There arcs two distributive laws in tho theory of acts. Law 1. This is expressed symbolically as: AV(BnC) = (A U B) n [A U C). In words, this is expressed as: (a) To form tho union of (B n C) with A, 13 and C arc 'cupped' individually with A, and the results are 'capped'; or (b) The union of A and the intersection of B with C is equiva- lent to the intersection of the union of A and B with the union of A and C. This is described by saying that 'union is distributive over intersection'. This is not true in ordinary algebra. Law 1 is illustrated in Fig. 20. Fio. 20 Law 2. This is expressed symbolically as: An{BuC) = (Ar\B}\j(Ar\ 0). In words, this is expressed as : (a) To find the intersection of A with the union of B and C, obtain the union of the intersection of A with B, and A with C. or (b) To find tho intersection of A with the union of B and C, 'cup' the 'cap' of A and B and the 'cap' of A and O. 26 In ordinary algebra, a x [b — c)=axb+axc. Law 2 is illustrated in Fig. 21. Fio. 21 4. de morgan's laws These laws deal with the complement of sots, and they have no counterpart in ordinary algebra. They are best illustrated by the use of Ye tin diagrams. Law 1. Thisstates: (A n B)' = A'\jB',i.e. (AB)' = A' + B r . (i) (AB)' = (A Pi J3)' is represented by the whole rectangle less the shaded portion (Fig. 22 (a)). -' ■ gama .- ■- \ =u ■'■ ^ y[\\\\ "■'-■"■'■'■ fb) (d> Fio. 22 (ii) A' + B' = A' U B' is represented by the portion un- Bhaded (Fig. 22 (b)). Therefore, {A n B)' = A' U B', or (AB)' = A' + B'. Law 2. This states: (A U B)'= A' O B', i.e. (A + B)' = A'B'. 27 (iii) (A U B}' = (A + B)' is represented by the shaded portion (Fig. 22(c)). (iv) A' C\ B' = A'B' is represented by the intersection of the shaded portions, which is the portion doubly shaded (Fig. 22(d)). Therefore {A U B)' = A' C\ B\ i.e. (A + B)' = A'B', 5. THE LAW OT ABSORPTION This Is best illustrated by means of a Venn diagram (Fig. 23). Fio. 23 It will be seen that the shaded portion represents A x B, or An B. This shaded portion is a subset of A. Therefore AB contains no element which is already a member of sot A. Therefore, A + AB m A. This is the law of absorption. It can be used when a term appears in an expression which repeats another term together with some further factors. Example. AC + ABC + AC = AC Here, the second and third terms repeat the first term with the additional factors Ti and 1 respective ly. It is said that the second and third terms are absorbed in the first. The law may also bo illustrated thus : Consider the expression a + ab Factorizing, = a(l + 6) Using 1+6 = 1, = a x 1 = a. Therefore, a + ab ■ — a. THE LAW. 1 ? OF BOOLEAN ALGEBRA Set algebra is a Boolean algebra, and the laws are listed below, in which A, B, 1, have the following meanings : A + B is the union of A and B, or A U B, AB is the intersection of A and B, or A f\ B, 28 1 is the universal set, or U, is the empty set, or 0. Any number system which has the following properties con- stitutes a Boolean algebra: 1. Commutative laws. A + B = B + A. AB =. BA. 2. Associative laws. A + (B + C) = (A + B) + C, A{BC) = (AB)C. 3. Distributive laws, (a) A{B + C) = AB + AC. (b) A +BG = {A + B)(A + C). 4. Union and intersection of a set with itself. A + A = A. AA = A. 5. Union of a set and its complement. A + A' = 1. G. Intersection of a set and its complement. A A' = 0. 7. Complement of the universal set is the empty set. 1' = 0. 8. Complement of the empty set is the universal set. 0' = 1. 9. The union of a set and the universal set is the universal set. I + A = 1. 10. The intersection of a set and the empty set is the empty set. a x A =0. 11. The union of a set and the empty set is the set. + A = A. 1 2. The intersection of a set and the universal set is the set. 1 x A ^ A. 13. De Morgan's Laws. (a) The complement of the union of A and B is the intersection of the complement of A and B. (A + B)' - A'B'. (b) The complement of the intersection of A and B is the union of the complements of A and B, (AB)' =A'+ B'. (c) The complement of the complement of A is A. (A')' - A. 14. The law of absorption. ,A + AB — A. NUMMARY Or THE LAWS OF BOOLEAN ALOEBKA 2. a + be - (a + 6)(o + c) 4. aa = a 6. aa' = 8. 0' = 1 29 1. a(b -r c) = ab + ac 3. a + a = a 5. a + a' = 1 7. 1' - 9. 1 + a = 1 10, x o = 11. + a = a 12, 1 x a = a 13. (a + b)' = a'b' 14. (<*)'- a' +6' IS. {a'Y = a. Examples of the use of the above laws 1. x{x' + y) = = xx' + xy = + xtj = ari/ 2. c{c + d) = cc + ed = c + ed = e 3. xy + xy' = x{y + y"i = as x 1 = a? /*• s(«' + t') = ss' + st' = = + at' = at' 5. au{a' + b') = aba' + abb' = + = 6. yz'(y + z) = yz'y + ya'z = y^' 4- = yz' 7. ab{ab + a'c) m aabb + aba' a = ab + = ab 8. (p + q')(p' + q) = pp' + pq + p'q' + qy' = + pq + p'q' + = pq + p'q' 9. pqr + pqr' + pqs - j»?{r + *") + JPST* = M x 1 + jtKjs = pq + pqs = pq 10. (x + y'){xy + x'y') <= xxy + xx'y' + y'xy + y'x'y' — xy + + + x'y' «= xy + x'y' 11. (xy' + yz)(xz' + y'z'\ = xy'xz' + xy'y'z' + yzxz' + yzy'z' = xy'z' + xy'z' + + = a^'s'. exeiicise 7 1. Given the following seta: A - {1, 2, 3, 4}, C - {3, 4, 5, 6, 7}, Find: 1. AKJB 2. 4. A Pi D 5. 7. A\J[Cn D) 8. ZJ = {3, 4, 5}, D = {6, 7, 8, 9}. inB 3. AUD iuoufl e. 4 n c n n in(CUD) 9. (A UC)n (.4 U D). ANSWERS 1. {1, 2, 3, 4, 3} 2. 3. {1, 2, 3, 4, 6, 7, 8, 9} 4. 5. {1,2,3,4,5, 6,7,8, 9} 6. 7. { } 8. 9. {1, 2, 3, 4, 6, 7}. 2. Given the following set€ : A — {Jack, John, Joan, Bill}, B — {Joan, Bill, Jim, Tony}, = {Jack, Joan, Henry). 30 {3, 4} { } { } {3.4} Find: 1. Af\B 3. (AnB)nc 5. Ar\(BC\C) 2. AuB 4. B n C 6. (AnB)KJ c. ANSWERS 1. {Joan, Bill} 2. {Jack, John, Joan, Bill, Tony} 3. {Joan} 4. {Joan} 5. {Joan} C. {Jack, Joan, Bill, Henry}. Find the solution set of integers for the following: 1. {.is | x + 2 = x] 2. {as | 3as + 7 = 22} 3. {x 1 3as + 5 = 2as + 5 + x] 4. {x \ x* = 25} 5. {x[ as>5}, where 17 = {1, 2, 3, 4, 5, 6, 7, 8}. ANSWERS {5} {+ 5, - 5} 1- { } 3. {all numbers} 5. {0, 7, 8}. List the members of the following sets: 1 . The set of even numbers between 10 and 20 divisible by 3. 2. The set of months of the year with names beginning with 31. 3. Tho set of days of the week with names beginning with T. 4. The set of all odd numbers between 7 and 9. ,"i. The .-ct- of vowels in the word 'iirithmotic'. (S. The set of all natural numbers between C and 12. 2. {March, May} 4. { } 6. {7, 8, 9, 10, 11}. ANSWERS 1. {12, *. 18} 3. {Tuesday, Thursday) 5. {a,e, i) Given the following sets: A = {a, b, c, d, e), P = {p, q, r}, N - {1, 2, 3), state whether the following are true or false: 1, »eP 2. 2cP 3. aeN 4. be A 5. q$P 6. 6JV 7. reP 8. 3eN 9. deA. ANSWERS 1. False 2. FalBe 3. False 4. True 5. False 6. False 7. True 8. True 31 9. True. 6. If x E U, and U is tho act of all the natural numbers, make a list of the following subsets of U: 1. {x\x + 2>5} 2. {as;3« - 1 = 5} 3. {x\ Sx - 1 = 2} 4, {x j As: = 5} 5. {a: | x + 6 = 6 + a:}. ANSWERS I. {4, S, C, . . .} 2. {2} 3. 4. {10} 5. U. 32 CHAPTER SEVEN THE GEOMETRY OF SETS Although there is no mathematical definition of a point, everyone has a good idea of what a point is. It has no size or shape, but merely position. However, a set of points can con- stitute a line. A lino is an infinite set of points, and any point on the line ia a member of the set. It is not possible to draw a straight line, but only a segment of the line. It is possible to show that there are as many points on a line one inch long as there are points on a line two inches long. Fto.24 Let AB (Fig. 24) be a lino one inch long, and let CD be a line two inches long. Let be any point in the plane of AB and CD. P is any point on AB. OP produced meets CD in Q. For every point such as P on AB there is a point such as Q on CD. Also, for every point such as Q on CD there is a point such as P on AB. This is an example of a one-to-one correspondence. On the two lines AB and CD, there are no points without a corresponding point on tho other. Not only is a line a set of points; a plane is also a set of points, A plane has only two dimensions, length and breadth. It has no thickness. Let (Fig. 25) be a point in a plane. Draw, in the plane, a number of points which arc a constant distance from 0. Those points will constitute members of the set of points which are equidistant from 0. Tho totality of this infinite sot of points will constitute a circle. This circle is called the locus of points equidistant from 0. 33 Fio. 25 U2HON AND 1NTEKSECTION" OB LETES 1. Lot it be required to draw a circle to pnas through the three given points X, Y, Z (Fig. 28). Let tho centre of this circle be 0. Then must be a member of tho set of points equidistant from X and Y, and must also be a member of the sot of points equidistant from Y and Z. Call these sots tho sots L and M, as indicated in the Figure. Then e L and € M. Therefore, is a member of the intersection of L and M, i.e., e L n M . In Fig. 26, X and Y are two fixed points. The set of points ■which are equidistant from X and Y constitute the perpendicu- lar bisector of XY. Fig. 26 Figure 27 shows two intersecting straight lines PQ and RS. Tho set of points which are equidistant from tho lines PQ and RS constitute the bisectors of the angles between PQ and RS, Actually there are two sets of points to bo considered, the set marked I, and the set marked m. Fio. 27 Fig. 28 2. Let it be required to draw a circlo to touch externally the three sides of a triangle. Let tho triangle be ABC (Fig. 29). Produce AG to D and AB to E. Then the circle will touch tho lines, CD, CB, BE. Its centre will be on the lines bisecting the angles BCD and CBE. Let th© bisecting lines be L and M as indicated in tho Figure. Then O e L and E Jlf. Therefore is a member of tho intersection of L and M . Fio. 2d 35 Therefore, E L n M. If AD is represented by D and BG by B, then MeBU Z>. If BE IB represented by E, then LeBV E. Therefore, e {B U £>) O (B U £). 36 CHAPTER EIGHT GROUPS Let a binary operation be denoted by the symbol *, and let a sot of elements be called G. Then tho set 6* will form a group if tho following rules, called axioms, aro obeyed. 1. If the sots X and Y belong to G, then X * Y is an element of G. This means that G is closed under the operation. In other words, the combination of two elements gives another element which is a member of the set 67. 2. The operation * is associative, so that X*(Y*Z)-{X*Y)*& This means that the order of combination of any three ele- ments does not affect the result. 3. must contain an identity element named I, such that X • I - I * X = X, for all X in G. This means that one element has the property that, if it be combined with any other element, tho element remains un- changed. 4. For every X in G tlicro must exist a unique element X', known as an inverse element, such that X * X' = X' * X = J. This means that every element of the set G possesses an inverse. A group which obeys the commutative law is called a com- mutative, or Abelian, group. This means that X * Y = Y * X for every pair of elements. THE GROUP STRUCTURE OF THE INTEGERS The following considerations show that the integers form a group : (a) The operation of addition is associative (b) The system contains a zero element (c) For every integer n, its negat ive —n is also in the system. 37 KINGS A system of elements is called a ring if there aro two binary operations defined for tho system having the following pro- perties : 1. Roth operations aro associative. 2. Tho system is an Abelian group with respect to one of the operations. This operation is denoted by the sign + > and is called addition. 3. The other operation is distributive with respect to addition. If this other operation is called multiplication and is repre- sented by . , then tho distributive law takes tho form: (a) a.(b + c) = a.b + a.c, and (b) (b + c).a — b.a + c.a. The distributive law must be stated in two parts (a) and (b) because the operation of multiplication need not be com- mutative. It will be seen from the above properties that the system of integers forms a ring. FIELDS A field is defined as a ring in which a unity element exists, and which has a reciprocal for o\'ery element except koto. It has a double group structure, ono for addition and one for subtraction. CHAPTER NINE TRANSFORMATIONS OR MOTION GEOMETRY A common property of the transformations to bo considered hero is the conservation of straight ness. This moans that straight lines in the 'object' will become straight lines in tho 'image'. There will also be conservation of distance. Thus the 'image' will be the same shape and size as the 'object', and the 'object' and its 'image' will be congruent figures. Distance -preserving properties are referred to aa isometrics, which means 'equal measures'. Such transformations aro sometimes called Rigid Motions. Since two translations, when performed in succession, can be equivalent to a third translation, translations can form a closed system, and so constitute a group. REFLECTION When an object is placed in front of a mirror, an image is formed behind the mirror, so that object and image are of the same shape and at the same distance from the mirror. There is lateral inversion, however, the imago being back to front, and left to right instead of right to left. Let the straight line PQ (Fig. 30) be rofleeted in the mirror AT, and let the image be P'Q'. P and P' are equidistant from XY. Also, P'Q' is the same length as PQ, and PQ and P'Q' are equally inclined to XY. 3S Fig. 30 (a) Reflection in the Coordinate Axes. Let the point P t = (x, y) be reflected in the axes (Fig. 31). Then the images will be P s ■ (-*, y) t P s m {-x, -y), P« m (*, -y). r §* - A . {-**) I 1 X * 1 1 _1 f 1. ■y) Fig, 31 (b) Reflection in the Line y = x. This is illustrated in Fig. 32. TIib point P t is reflected in the line y = x, and produces an image at P a such that P,M - P t M. Tho triangles OPiilf and OP t M wiU be congruent, so that 0P t = OP s , and angle P x OM = angle P s OM. Therefore triangles OP x A and OP,B will be congruent. Therefore OB = OA = x and BP t = -AP lP Therefore P 2 will be the point {y t x). Therefore, on reflection in the line y = x, the coordinates of a point are interchanged. Y B -A /\ yY=* I MX if- / AX Fio. 32 40 (c) ifrjfeciwn i» tfie .Lwie a; + y = 0. It may be shown that in this case the point (a^, y x ) becomes the point (— y v -a?!). Thus the coordinates will be interchanged and the signs will be changed. As a reflection reverses the sense of irregular shapes, no trans- lation or rotation or combination of them can have the same effect, although both a translation and a rotation can be pro- duced by successive reflections. Thus, reflections may be con- sidered as basic isometries, because all other isometries can be obtained by suitable combinations of reflections. ROTATION This is a common isometry and occurs wherever there are wheels. Every line of a rotating figure turns through the same angle. In order to specify a rotation it is necessary to state the centre about which all points are rotated, the angle of rotation, and the direction of rotation. It will bo considered here that all points will remain in the Bame plane. The centre of rotation will be tho only fixed point. The convention adopted in geo- metry is that an anticlockwise turn is positive. However, bear- ings are measured positively from tho North in a clockwise direction. An important rotation is the half turn, a rotation through ISO 3 , or n radians. This causes the signs of coordinates to be reversed (Fig. 33). (-*&) Fig. 33 Figure 34 illustrates a quarter turn, and transforms tho point P3 ■ (x, y) into the point P t = ( — y, x). 41 p. r {-Y<*J\~ -§?" ^^ 1 M X Fio. 34 BOTATIONAL, SYMMETRY This kind of symmetry is frequently met in nature. Figure 35 shows the mystic pentagram. P v P t , P s , P t , P t are equidiatantly placed around the circumference of a circle. They are produced by five anticlockwise rotations of = 72° around the centre 5 0. The points are carried on to themselves by five reflections in the lines joining to each point. Fio. 35 Tho ton operations of reflection and rotation form a group, the group of self- transformations of the pentagram. The best known examples of rotational symmetry are snow- flakes winch possess hexagonal symmetry. thaxslatio:* In this kind of transformation all points in the plane are moved equal distances in the same direction. The effect in any 42 particular figure is such that it remains the same way up. It could in some respects act as an introduction to vectors. A good example of translation could be the flight of a squadron of aircraft in formation. Figure 36 shows the effect of a trans- lation on the line P t Qi. It becomes tho line P 2 <2 a . All pointa on the lino are moved the same distances parallel to each axis. Let the x coordinate be moved a distance h, and let the y coordinate be moved a distanoo it. Then each point (x, y) on the line P-^Qi will become the point [x + h, y + k) on the line F:o. 36 SHEARING Shear translation is a basic transformation in that group of transformations called affinities. If a reversible linear mapping is followed by a translation, the product is an aifine transfor- mation. The system of such products is an afline group. In a shear translation each point moves parallel to a given line and tho distance moved by each point is proportional to the distance X .1 J O) (b) Fia. 37 of the point from tho line. Shearing transformation conserves the area, or tho volume, of the figure transformed. This can bo seen in the case of a pile of books (Fig. 37). If the pile is 43 subj ected to a shearing forco the sti ape is changed from that shown in Fig. 37(a) to that shown in Fig. 37(b). It will be seen that the total area of the end is the same, and the total volume of books is unaltered. However, shearing is not a shape -preserving transformation. SIMILARITY Similarity, or dilatation, or enlargement is a transformation which preserves shape. That is, shape is invariant. This is the kind of transformation obtained when a photograph is 'en- larged*. This invariant property of similar right-angled triangles Fig. 33 is used in trigonometry. A dilatation may be defined as a trans- formation in which a point P and its transform P t are in line with a fixed point 0, such that 0P l = 7. . OP, A being a constant, called the dilatation constant. Figure 38 shows triangle PQR transformed under dilatation into triangle PjQjR^ 44 CHAPTER TEN TOPOLOGY This is sometimes called the rubber sheet geometry. It is the study of the properties of shapes which are invariant under a topological transformation. All the shapes in Fig. 39 or© tho Fia. 39 same topological ly, the invariant property which they all pos- sess is that they all have an inside and an outside. If each shape wore drawn on a sheet of rubber, any one could, by suitable stretching, be transformed into any other. One condition is that there should be no tearing of the rubber. This means that any two points which are neighbours beforo the transformation will remain so afterwards, and any two points not neighbours before wOl not be neighbours after tho transformation. The shapes are said to be topologically equivalent. The shapes of Fig. 40 are also topologically equivalent. oo O <x> B Fio. 40 In the same way, the solid figures, sphere, cone, cylinder, and pyramid, are all topologically equivalent. They all have an inside and an outside. Just like ordinary geometry, topology concerns linos, points, and figures, except that in topology they are allowed to change their size and their shape. Topology deals with position rather than with size and shape. It deals with the properties of figures which remain the same howover the figures are stretched or bent. There is no meaning for distance in topology. If two points are one foot apart, this could be stretched to two feet, or three feet and so on. Similarly, the straightness, or curva- ture, of the line has no meaning; neither has the size of angles. 45 All these can be changed without affecting a problem. Thus, in topology, there are no rigid bodies. Lines can stretch or bend, and plane and solid figures can change their shape. However, lines, and plane and solid figures are continuous. A lino has no holes or gaps. Fig. 41 Figure 41 shows two points X and Y. They are joined by a line XY called the arc XY. The line X Y may be made curved and longer, but providing it is not made to cross itself it is always the path from X to Y. The shapes of Fig. 39, namely the triangle, circle, quadri- lateral, and irregular closed curves are all topologically the same. Each is called a closed circuit, or simple closed curve. Fro. 42 Figure 42 shows just one of these figures; but they can all bo treated alike. O is a point outaide, and P is a point inside. However the figure be distorted, the line OP will always cross an arc of the curve. This is because there aro no gaps in the contour lino; it is a result of the principle of continuity. Thus a closed curve will divide the plane into two parts, an insido and an outside. Since it is always necessary to cross the boun- dary line in order to pass from the inside to the outside, the action is said to be invariant. A situation which is invariant is one which docs not change. Although it is possible to change the shape of a lino or a figure by bending or stretching, theso form topological transformations, but they do not alter the topological nature of the line or figure. In order to maintain the topological nature of a line or figure it is necessary that it be neither cut, torn, nor folded. If a closed curve is cut or torn or folded a new topological figure is formed. Thus a line or curve has another topological invariant, and that is the order of the 46 points on it. If P, Q, R, and 5 are consecutive points on a line or figure, then they will remain in that order under a topo- logical transformation. A topological transformation always produces equivalent figures, so that a circle, square, quadri- lateral, ellipse, triangle are all equivalent. If the ends of a line are joined, or if a closed curve is cut, new figures are formed. Fio. 43 Example. Figure 43 shows a strip of paper which is marked on both sides with a continuous pattern of oriented circles. Form this into (a) a cylinder and (b) a Moebius band (see p. 51). Is it possible to find a point on either band at which the sense of rotation of two adjacent circles suddenly changes? It will be possible to find such a point M on the Moebius band, and the surface is said to be n on -orien table. There will be no Bueh point on the cylinder, which is said to have an orientable surface. CLASSIFICATION OF TOPOLOGICAL PIGCEES A sphere, a cube, and a pyramid are topologically equivalent. They are continuous surfaces, and divide space into two regions, un insido and an outside. They are called simple closed surfaces. Eaefa can be changed into tho other by simple distortion. In topology, surfaces or figures are classified by tho number i >f cuts needed to simplify the surface or figure. Figure 44 shows (a) (b) Fig. 44 a disc-like figure with a hole, similar to an annulus. It divides the plane into three regions, P, Q, and E. If Fig. 44(a) is cut 47 as shown in Fig. 44(b), a simple closed curve is obtained, in which P and B aro both on the outside. Three dimensional figures are classified in the same way, by finding how many cuts are needed to change the object into a simpler closed surface like a sphere. (a) W FlO. 43 Figure 45 shows a torus, a doughnut-ahaped solid. A single cut will produce Fig. 45(b), which is a simple closed surface. As a torus and a sphere are topologically different, they are said to be different in connectivity. The hole in the torus is actually outside the body, not inside it. A sheet of paper is said to have two surfaces and one edge. A cylinder which is open at both ends has two surfaces and two edges. The inner tubo of a motor car wheel, like a torus, has two surfaces but no edges. It is possible to classify shapes by counting the oross-cuts needed without dividing it into more than one piece. A cross-cut should begin and end on an edge. If a sheet of paper is cut from edge to edge two sheets are formed, but if an open cylinder is so cut, only one sheet is pro- duced because it can then be opened up into a single flat sheet. This means that a sheet of paper is a simple surface, and the cylinder is a singly connected surface. A sphere is also a singly connected surface because it is possible, although difficult, to make a hole in the surface and stretch it into a flat sheet. ecxeb's tqkmola This is a formula concerning regular polyhedra. A polyhedron is a solid whose faces consist of a number of polygons. The polygons may be a triangle, a quadrilateral, a pentagon, or a hexagon. Figure 46 shows regular polyhedra. All the faces are congruent figures, and all the angles at the vertices are equal. 48 (d) FlO. 46. (a) Tetrahedron (b) Hexahedron (c) Octahedron (d) Icosahedron (e) Dodecahedron In a simple polyhedron, let the number of vertices = V, the number of edges = E, and the number of faces = F. Then Euler's formula states: V - M +* - 2. This is a topological formula, because it concerns only the numbers of the vertices, edges, and faces, and not the sizes of them, or the lengths, areas, or straightness of the parts. EULER S NETWORKS Leonard Euler, the Swiss mathematician, was a pioneer in the study of networks. His researches into these problems were started by the Koenigsburg bridgo problem. Koenigsburg was the name of a Russian town. A river flowed through this town, ■find two islands in the middle of the river were joined to the banks by seven bridges, as shown in Fig. 47. The problem was, ■'■ -Y'V.^ ■'■■: y ... ■ :.. ■■• ..: , / ■ . '■,■■■-- ■■ ,.'..''■.-'■'*■■"-'■ . . Fig. 47 ■ ■--•■•'•. ■■■.<■:&»■:■■ - 1 ,i:r:...J ;■.:?;:"::■.,;■' . . ■..■•'.;: .... "Is it possible to cross each of the bridges, and cross each bridge once only I' Euler discovered that it was not possible to cross each bridge without recrossing any of them. Euler tackled the problem by drawing a network P, Q, R, S, as in Fig. 48. In this network, tho vertices are P, Q, R, and S. These are points where the lines cross. The joining lines are called axes. Since the number of arcs at the vertex P is 3, P is then an odd vertex. Q has live arcs, ami so it is also an odd vertex. Euler discovered the following rule : If the number of vertices at which an odd number of arcs meet is two or less, then it is possible to draw tho figure in one continuous pencil Era. .($ movement without drawing any part more than once. If there are more than two vertices at which an odd number of ares meet, then sueh a drawing is not possible. 50 / ^ O > —• " 1 * < Euler also discovered that any network with only even ver- tices could bo traversed in one journey. He also discovered that if a network contained two and only two odd vertices it could be traversed in one journey, but it would not be possible to return to tho starting point. It would be necessary to start at one odd vertex and finish at the other odd vertex. He also discovered that if a network contained four or a higher even number of odd vertices it would not be possible to traverse the network in one journey. THE MOEBIUS STRIP In tho eighteenth century the mathematician lloebius dis- covered that there were surfaces with only one side. The sim- plest is the Moebiua band. This is just a rectangular strip of paper with tho two ends joined together after making a half twist (Fig. 49). Fio. 49 If an insect crawled along the middle lino of this strip, it would return to its original position upside down, and it could walk from any point to any other point, on either side, without crossing an edge. If a painter started to paint the outside of the band, he would paint the inside as well. the KLEIN BOTTLE This is a one-sided surface named after the German mathe- matician Felix Klein. It is a single surface with no inside, 61 outside, or edges (Fig. 50). It is formed by drawing the smaller end of a tapering tube through one side of the tube and then enlarging the former until it fits the latter. Fig. 50 THE MAP COLOUR PROBLEM This is a famous unsolved problem in mathematics. Maps are often drawn with countries, which havo a common border, coloured differently. It has been found, so far, that only four different colours are needed for this purpose. No map has yet been conceived whose colouring requires more than four colours. It has never been proved that no more than four different colours will ever be necessary. has shown that it is not possible to connect P, Q, and R with W, G, and E and still fulfil the authority's requirement. (b) This is the problem of the Caliph's daughters. They had so many suitors that the Caliph set each one a problem, which was 1 1 1 / i ©- / / -©J ©J / KD— Fio. 52 to connect the two sets of figures, 1, 2, 3, shown in Fig. 52, so that the same figures are joined without the connecting lines crossing each other or any lines in the Figure. This is a topo- logical problem, and it has no solution. TFTREE-TO -THREE COXXliCTIOSS (a) Three houses P, Q, and R aro to be supplied with water, gas and electricity. A local authority requirement is that pipes and wires must not cross each other (Fig. 51). Experiment "1 p 43 R — j ^^pL — i ! V"--.. K N **• S ■^ ** V •* \ s i <s> | i_. I^J Fio. 51 52 EXERCISE 8 1. State which of the diagrams in Fig. 53 are topological lines. Q (a) J — L (c) (b! n &^ui Id Fio. 53 2. State which of the diagrams shown in Fig. 54 are simple closed curves. T D (a> (b> : (c) €•*> Fig, 54 3. For each of the networks given in Fig. 55, state the number of even and odd vertices, and investigate whether the network can be traversed. r^ CO (e) Fia. 55 4. Classify the following surfaces regarding their connectivity: 1, a tennis ball 2. a hosepipe 3. an inner tube 4. a coat 5. a plate. 5. Make a Moebius band which contains two half-twists. Mark it and cut it along the middle. Find out whether the surface is 54 (a) one sided or two sided and (b) orientable or non -orientable. C. Show that the least number of intersections possible with four houses and throe supply stations is three. 7. An area of tessellated flooring is completely covered with regular hexagons. How many colours are necessary to ensure that adjacent hexagons are of different colour? 8. Find the minimum number of trips needed to trace this net- work given in Fig. 56. Fig. 56 ASSWEBS TO EXERCISE 8 1. (a), (o) 2. (c), (d) Network Even vertices Odd vertices Traversable (a) 2 2 yea (b) 6 DO (0) 4 yes (d) 10 yes m 4 S no (2) singly connected (4) doubly connected (1) singly connected (3) doubly connected (5) simply connected (a) two sided, (b) orientable Three 8. One 1TM— E 55 CHAPTER ELEYEX THE ALGEBRA OP CIRCUITS Tho algebra of logic has a direct application to the design of electrical circuits. However, in that which follows, no mention will be made of voltage, current, power, or resistance. The chapter will be concerned only with tho result of inserting switches into circuits. Sections of circuits will be denoted by the small letters, a, b, c, etc. If section a has a current flowing through it, this fact will be expressed as a = 1. If thero is no current flowing through the section a, then this will be expressed as a = 0. Diagrammatically, these situations are illustrated in Fig. 57. SWITCH CLOSED Fig. 57 ^o SWITCH OPENED It must be noted that the and tho 1 are not numbers in these situations. and I are the only symbols needed, because a switch can be either open or closed, and nothing else. (i) Switches in Series. Figure 58 shows two switches in series. ■^ Fig. 58 A current will flow only when both switches are closed. This situation is denoted by ab. (ii) Switches in Parallel. In this situation (Fig. 59), current will 56 Fio. 59 flow when either a or b is closed, or both are closed. This is represented by a + 6. TABLES (iii) The Multiplication Table. This follows from (i) above. a b ab 1 1 1 1 1 (iv) The Addition Table. This follows from (ii) above. a b a + b 1 1 I 1 1 1 1 (v) Series Parallel Circuit (a). Figure 80 shows circuit a in series Fio. 60 with b and e in parallel. This is represented by o(6 + c). The following table shows the result of various combinations of switches open or closed. 57 a b 8 o(6 + e) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^ ct = 1 o'=0 Fio. 62 *X. (vi) Series Parallel Circuit (6). Figure 61 shows series circuit a and b in parallel with series circuit a and c. The whole circuit will be represented by ab + ac. Fio. 61 ocQ cr'sl It will be seen that there is no intermediate position, a' means that the state of wire a' ia opposite to the state of wire a. a' means that the switch is open when a is closed and closed when a is open. Example. The diagrammatic representation of o + a f is shown in Fig. 63(a) or (b). There are two possibilities. The following table shows the result of various combinations of switches open or closed. a 6 c ab + ac 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 TWO-WAY SWITCHES Some switches are arranged in such a way that in one posi- tion current flows through one circuit and in the other position current flows through another circuit. This is illustrated in Fig. 62. 58 Fio. 63 The addition table for this arrangement is ; a o' a + a' 1 1 1 1 59 Example. The symbolic representation of the circuit in Fig. 64 ia aa' = Q. — — <»■ ^fc <? a' Fiq. 64 Example, The symbolic represent at ion of the circuit in Fig. 65 ia (a -i- b){a + c) = a + be. Example. The diagrammatic representation of the formula o + fee is given in Fig. 66(a) or (b). There are two equivalent circuits since a + be = (a -f b){a — cj. ~d a*bc{o'6Ho*c) (M Fro. 66 60 The table for this circuit is s * c a + be 1 1 1 1 1 1 1 1 1 l 1 1 Example. The circuit of Fig. 67 may be represented by the formula {ab + c)(ad + c'). Fig. 67 However, (ab + c)(a6 + o') » ab + cc' (Distributive law) = a* -J- = afe. Therefore the circuit of Fig. 67 may be simplified to the circuit of Fig. 68. ^ *<. Fio. 68 Example. By the associative law for addition, a + (fe + c) = (O -r b) + C. This means that the circuits in Fig. 69(a) and (b) are equivalent. Fio. 69 61 BOOLEAN' ALGEBRA AND SWITCHING CIRCUITS It will be seen from the foregoing that the laws of Boolean algebra can be applied to switching circuits, in the same way as the laws can be applied to sets. A structure which behaves in this way is said to bo polyvalent. Boolean algebra can be used in the simplification of electronic cirouits, as will now bo Atoms* Consider the circuit indicated in Fig. 70. Fig. 70 This circuit consists of five parallel circuits, each made up of two switches in series. It will bo seen that this can be ex- pressed in Boolean algebra by tho formula pq + pr + pa + qr + qa. The expression may be factorized into p{q + r + a) + q(r + a). This expression represents two parallel circuits (1) p(q + r + a) and (2) q(r + a). (I) contains p in series with q + r + a in parallel and (2) con- tains q in series with r + a in parallel. Therefore it can be represented diagrammatically as in Fig. 71. This circuit (Fig. 71) will have the same switching effect as the circuit in Fig. 70. However, there is another arrangement. Since M+P r +J >s +gr + ga=j>g + {j)+ g){r + s), the original circuit may be arranged according to tho expression pq + (j=> t q)(r + a) which is indicated in Fig. 72, Fio. 72 exercise: 9 1. Write down the Boolean function for tho oircuits shown in Fig. 73, simplifying the expressions where possible. 2. Expand the following expressions and simplify where pos- sible: (i) a{a + 6) (ii) ab'{a + b) (iii) (o 4 &')(«' + 6) 3. Factorize and simplify the following: (i) ab + ac (ii) a' + a'b (iii) ab + be + be' (iv) ab + c (v) ab + a'b' ANSWERS TO EXERCISE 9 1. (i) a. a = a (ii) a + a = a (iii) a + ab = a (iv) (a + b){a + c) = o + 6c (v) (a' + &')(«' + b)(a 4- b) = a'b (vi) (a + a) (a + b)[a + c) = a (vii) a + b(a' + c) = a + b (viii) ab + a'(b + c'){b' + c) 2. (i) a + ab = a (ii) ab' (iii) ab + a'b' 3. (i) o(6 + c) (u) a'{l + 6) = a' (iii) 6 (iv) (a + c){6 + c) (v) (a + 6')(a.' + 6). G3 V "o \ < > to CHAPTER TWELVE DETERMINANTS A determinant is an arrangement of letters or numbers set out in equal numbers of rows and columns, and the arrangement is placed between two vertical straight lines. Here are some examples : a h g 3 5 h b f , 4 2 3 2 7 4 6 9 2 8 6 g f c A determinant with two rows and two columns is called a determinant of the second order, and one with three rows and three columns is a determinant of the third order. a b The symbol Btands for ad — 6c . c d la b c The symbol p q r stands for x y z 3 r p r - b + c G4 y s \x z is called the minor of a, is called the minor of b, and is called the minor of c. P 2; x y q r y « \p r X 2 P ? x y The co -factor of an element is the minor of that element with the proper sign attached. 65 Example The sign of a co-factor is positive or negative according as the sum of the number of the row and the number of the column struck out to form the minor is even or odd. 3 5 = 3 x 2 - 4 x 5 = 6 - 20 = - 14. 4 2 7 | 4 9 4 6 2 5 2 8 = 3(6 x 5 - 8 x 9) - 2(4 x 5 - 2 x 9) + 7(4 x 8 - 2 x 6) = 3 x -42 - 2 x 2 + 7 x 20 m -126 - 4 + 140 - 10. RUUE OF SAKRtTS This is a rule for finding the value of a determinant. The first two columns of the determinant aro repeated on the right of the determinant, as shown in Fig. 74. The expansion of 3 2 7 16 9 Example 4 6 9 2 8 5 8 5; -pfe ^: *ofr -qga *6gp -reb Fig. 74 *ceq the determinant is written down by taking the algebraic sum of the products formed by the elements on each of the Bis diagonals shown, the products taken downwards being positive and those taken upwards being negative. The determinant may also be evaluated by repeating the first two rows below the determinant and multiplying as before. 3 2 7 Example. Use (he rule of Sarrus to evaluate 4 6 9 as shown in Fig. 75. 2 8 5 -84 -aie j h 3 s 2 - , 7 -/ A' J* *■ J** -. ■*■ ^ ■*■ ' r "** - »* N r- 4 \6„' >„' '""jt * 6 ,"' >< >'. e ' e' 5 ' "2 ~e ^ ^ *•% ** ♦ 90 *36 -40 Fio. 75 ♦ 224 Value = 90 + 36 + 224 - 84 - 216 - 40 = 350 - 340 = 10. SOLUTION OF EQUATIONS BY DETEB3HNA>'TS Consider the equations OjX + o,j/ = c L aj* + b^y = c s . It may be shown that their solution is given by c t b t — <?,&! y = a^.. - «,c, a-yb^ — a^by Cl K Cj ^ <h h <>i b* <h c, a* c, a, b t I <h & s Thus the roots of the two simultaneous equations may be expressed in determinant form. Example. Solve the equations 8x + y = 3 to + %y = -1. The solutions are : x = 3 1 Li 2 8 1 9 2 GO 6 + I _ 7 _ 16 - 9 ~ 7 ~ 67 8 3 9 -1 8 I 9 2 in which, -8 -27 _ -35 ~lti ^9~ ~ ~1~ -5. If there are three equations in three unknowns, such as: a t x ■+• b x y + c L « = d^ Oj« + &# + Cjt = d 3 a s a; + b$ -f CjS = d, The solution is x = d, b, Cl d, &> Cl d, 6 3 c» a i ''i Ci «» &h Cl «s &* c» a t &i d, *i *>» d, a. 6* <*» "j &i Cl y = "l rfi c L »2 d 2 * <*» <*. Ct Ol bi <h a 2 b. C| Oj h C., <h b t The denominator, in each case, is a third order determinant, and it is usually denoted by the symbol A- Then x = ^i, y = ^, z = A», A " A A 68 Ai- di 6 t C, I I d s J> 2 c, d s 6 3 c 3 % o t d L . a, 6, d 2 a» b 3 dj [ A 5 = A = a. d 2 Cj a 3 d, e s a, b t c, ^j 6 2 c 2 ' °3 6 S C 3 PHOFEBTIES OF DETERMINANTS 1 . The value of a determinant is unaltered if rows are changed to columns «i &i Ci a t a, a, a, &» c* = ! &i b t 6, 2. If two rows (or two columns) are interchanged, the sign of the determinant is changed. o s &, c, J <H *i «i «l h Ci a 3 b. c t «., b 3 %j a, 6j c, | 3. If two rows (or two columns) of a determinant are the same, the value of the determinant is zero. a i b h c v a i b t Cy ' = <h b t o,| 4. If the elements of any row (or any column) of a determinant be each multiplied by the same factor, the result is the product of that factor and the original determinant. 69 fat! *6 1 kc v o, &i c, rtj. fc 2 c a -ft a. b* H «j 6* «3 % b 3 c 3 5. If each element in any row (or any column) constats of t ho euro of two terms, the determinant can be expressed as the sum of two determinants. °1 ■+■ ""l bl c l . "i ''i ''i Xj 6, c % a 2 + #2 fcj c 8 = 0; Ilj Cj + x t K c 2 a, + as,, b 3 c 3 a 3 b 3 c 3 #3 &3 «3 G. If the elements of any row (or any column) are increased or decreased by equal multiples of the corresponding elements of one or more of the other rows (or columns), the value of the determinant is unaltered. a x + kb t &! c t a t + kb a b t c t a 3 + ** s 6 3 c, EXERCISE 10 Evaluate the following determinants: a% &i Cj Oj 6„ c. a s & 3 c 3 i. 2 3 4 5 2 5 4 7 3 6 5 8 9 J 2 -2 -1 6 1 -1 4 3 5 I 5 -2 3 1 5 1-2 4 6 3 3 8 4 -6 3 12 16 8. 4 -3 2 i 4 -1 70 9 -7 4 8| 3 8 7 2 4 -5 1 3 -4 -3 2 7 -31 5-9 2 Use determinants to solve the following equations 10. 2x + y = 4, 3a; + 4y = 1 11. 5a; + 3y = -6, 3a: + By = - 18 12. 5a> + 2y = 2, 3x - By = 26 13. 3x - 2y = 6, 5a; - y = -4 14. 2k - 5y = 10, 14a; + 5y = 6 15. 4a; - Zy + z = 7, 3a + 7y - 3a : ■■ 8, 16. 2a: - 3y -r 5z = 4, 3a; + 2y + 2a - 3, 4a; + 1/ — 4s = —6 17. 2a; + 3y + a = 6, 2a; + 9y + 3s = 14, 4as + y + 2a = 7 18. 7a; - 4y + 2a - 4, 2a; + 3y - "2 = -5, 5a: — 2y — a m — 1. ANSWERS TO EXERCISES 10 1. -2 2. 11 3. -33 4. -43 5. -35 6. -684 7. 70 8. 178 9. 10 x = 3, y = -2 12. a; = 2, y = -4 13. a: = -2, y = -6 14. s = 1, y = -f 15. a: = 2, y = -1, z = -8 15. * - -i, y - |, S - I IT. x = 1, y - 1, z = 1 18. a; = 2, y = 4, 2 = 3, 5a: — y + 2a = 5 11. a; - 5, y = -f 71 CHAFTEH THIRTEEN MATRICES A matrix is a rectangular arrangement, or array, of elements. The elements may, or may not, be numbers. A matrix may be regarded as a code of instructions. It can have any number of rows and any number of columns. When describing a matrix, the number of rows is placed first. Thus a 3 x 4 matrix will contain three rows and four columns. Matrices are arrange- ments such as : 4 -1 I 1 -4 -1 [a b c], *J °1- -1J The elements arc enclosed in brackets to distinguish them from determinants; it is possible for a matrix to have only one element. Matrices have their uses in many branches of mathe- matics; in particular, they are useful in the study of reflection, translation, rotation and other distortions of geometric figures. A matrix has no numerical value, because its solo purpose is to indicate an arrangement of elements, although the ele- ments could bo numbers. A square matrix is an arrangement having an equal number of rows and columns. Such a matrix must not be regarded as a determinant, which is a number ob- tained by multiplying, adding, and subtracting the elements in a certain manner. Although rows and columns can be inter- changed in a determinant, they cannot bo interchanged in a matrix without altering it. If a matrix has only one row, it is known as a row matrix. A matrix which has only one column is called a column matrix. To save space in printing, a column matrix is often written within braces as {x lt z», x s , a - ,,}. A single number could be regarded as a 1 x 1 matrix, that is, as a matrix with only one element. Also to save space, a matrix is sometimes denoted by a capital letter, or a small letter in Clarendon type. 72 MULTIPLICATION Or MATRICES 'a 6~| fp q | and o d (a) Let be two matrices. Then '1 P and u L [a 6~[ fp q~\ Vap -f- br aq + 6s" c dj [r sj [cp + dr cq + eie_ (b) Two matrices can only he multiplied if the number of rows in one is equal to the number of columns in the other, they are then said to bo compatible. (i) Let [a 61 \p~ and [c dj !_?. [c d] be two matrices. Then (ii) Let 'a b' c d .e / a b' c d L. /J and \_r s. p q _r s_ (iii) Let "N 3?1 £*2 "^3 and & a Ui 3/j yJ -V 3F i 2 . V Lfr» 'ap + bq' _cp + dq_ be two matrices. Then ap + br aq + 6s" cp -I- dr cq -r ds ■ &P +fr eq + fa. be two matrices. Then xji 1 + asj&j + xj> z ADDITION OF MATRICES Matrices arc added by adding corresponding elements. Thus '«i &i fi c, x 1 y 1 z 1 "Oj + x t 6j + jfe c t + z t ' . x i Vi z i\ L°2 + *£ &° + Vi C, + z z SUBTRACTION OP MATRICES Matrices are subtracted by subtracting corresponding elements. a, - *i b t - p 1 o, - z; a t - x t & s - j/ 2 c 2 - % MULTIPLYING A MATRIX BY A FACTOR This is defined by the following: Thus *°i 6 i c f ~*t yi *r _a 2 fc 2 Cj_ J*2 y 2 H. "a b~ Ka Sb' c cJ_ JSe Kd_ EQUAL MATRICES Two matrices with numbers as elements arc said to he equal if, and only if, they have tho same number of elements arranged in exactly the same pattern and have the same numbers in the same places. The sign => means 'implies* and the sign o means 'implies and is implied by'. Therefore: oa=p,b=q,c=r,d = <&a = p,b =q,c = r, d = s, e = t,f = u. THE UNIT MATRIX This is denoted by I and then "1 I = ~a b ~P 3" e= oo = _C d_ jr 8_ "a b c" "p q r~ d e /. s t M_ o i. 74 When the matrix 1 0" 1 multiplies another matrix, the 'I 0*1 second matrix remains unchanged. The matrix 1 , there- fore, behaves like the natural number 1. That is why it is called the unit matrix. It is also sometimes called the neutral matrix or tho identity matrix. It can have other forms, depending "1 0' upon the order. Thus I may also be the matrix 10 .0 I. which is a 3 x 3 matrix. If a is a square matrix and I is a unit matrix of the same order, then I.a = a.l = a. THE ZERO MATRIX Any matrix whose elements are all zero is called a zero matrix. The following are all zero matrices : ■Q 0" "0 0" -<r .0 o. I _0 0_ 1 iA The zero matrix is sometimes denoted by 0. If A is the matrix c d .then, A.O = 0.A The zero matrix is sometimes called tho null matrix. A DIAGONAL MATRIX This is a square matrix whose elements are all zero except To 0"! those in the leading diagonal. For example, diagonal matrix. A unit matrix is thus a diagonal matrix. 75 6 L0 c. is a THE INVERSE MATRIX Let one matrix bo A, and let a second matrix be B. If A . B = I, then the matrix B is the inverse of matrix A. There- fore, A.B = B.A = I. If £ is written as X -1 , A- 1 = 4 _I . A = I. METHOD OF BTNDrNQ THE INVERSE OF A 2 X 2 MATRIX ~a b~ Let A be the matrix c d A A — c o "a" A - 3 11 n -S - 11 7 11- Then the inverse of A = A -1 = where A = ad ~ be, and called the determinant of the matrix. "7 2" Example. Find the inverse of A = .5 3. Here, A = 3 x 7 - 6 X 2 - 21 - 10 = 11. Now, interchange the 3 and the 7, change the signs of the 2 and the 5, and divide each element by 11. Then^l- 1 = '2 3~ Example, Find the inverse of A = .5 »_ Here, A=2x9-6x3 = 18-15 = 3. Interchange the 2 and the 9, change the signs of the 5 and the 3, and divide each element by i = 3. Then A- 1 SIMULTANEOUS EQUATIONS USETG MATRICES Consider the equations a,a -f b^y = c x a^ps 4- b 3 y = e,. 76 r * -*- p -11 3 3 •-S 2 -s 2 - S i- _ 3 U These may be written symbolically as ; "a, &i" Let the matrix be A-K Then .Osi 6J IS. a i b i }- «s 6 i. Li/J be called ^1, so its inverse will ~ c i" Multiplying throughout by A~ l "<: r c 'i A- ■ l .A m A- 1 LyJ L«J V rvi -'- -:'. = A- 1 A. From this, as and y may be determined. Example. Solve the equations 3« + y =5 &s + 2y = 8. These may be written in matrix form as: 3 1" 5 2. _.'/. (1) A = 3 1' 6 2, Therefore, A~ l = ■1 1" 5 3. Multiply (1) throughout by 2 -r -5 3 ■3 r X U/J 2 -1" 3 -1 -5 3 J _^ r 2 -n -5 77 But 2 -1" ■5 3. 2x5-1x8' .-5x5 + 3x8 10 -8" 1:1. -M -1 + 24_ -1 .'. x — 2, y — — h Example. Solve the equations ICte + 4j/ = 12 7* + 3?/ = 8 ■ [10 41 V| •121 These may be written aa ^7 3. J/_ = 8 - " A = "10 4" . 7 3. . Then A -i = I 9 7 i -<- i 10 3T- ! -7 - 2 — 2" 5 (I) Multiplying both aides of (1) by A~*- 1- J -2" "10 41 n*i r i --.r [12] 1 2 -7 . 2 5 . ? 3_ jf. h 5 J _ 8_ V r ? -2"i "12" Jl- -7 _ 1 5 L sj But r 3 —2' i 1 5 . -[- .'. ar = 2, y = -2. x 12 + (-2) x 8' x 12 + 5 x 8 18 + (-16)" 42 + 40 2" -2 EXERCISE 11 Use matrices to solve the following equations : 1. x + 3y = 5 2. 5* + 2y = H 2z+y = 9 3a: + y = 6 78 3. 4x + y = 9 3x + 4y = 10 5. as + 2y = 10 x + i/ = 6 ANSWERS TO EXERCISE~11 1. 2, 1 2. 1, 3 4. 5, 1 5. 2, 4 4. s - 3y ="2 5x + y = 26 6. a: + 3i/ = 1 3k + y = —5. 3. 2, 1 6. -2, 1. APPLICATIONS OF MATRICES 1. An anticlockwise rotation of 90° about the origin. Let the point {x, y) be transformed to the point {x lt j/ v ) {Fig. 70). Since trianglea P OM l and PfiM t are congruent, then x Y = -y y x - * Fig. 76 These equations may be written as x l = O.a: — \.y y l = l.x + O.y. These may be written in matrix form as ~ x \\ r° — n r* Therefore the matrix describing this particular transforma- tion is _ -1" .1 0. 79 2, Reflection in the x-axis. Let the point (as, y) be transformed tinder reflection in the as-axis to the point (as,, y t ) (Fig. 77). Fio. 77 I Y p Q ]u.y) O A 1 p. It will be seen that as, = x Vi= - ■y- These equations may be -written as *, = 1. x + O.y ff , = 0. as - \,y. Those may bo written in matrix form as ~*i r^ "1 i n Ufil Lo -ij LsJ Therefore the matrix describing this particular transforma- tion is r i .0 -lj 3, Enlargement. Let tho vector GP be stretched to become the vector OP lr such that Of, = K.OP a (Fig. 78). Then and as, = Kx Vi - Ky. Y f\ y St**** jSvt.A X Fio. 78 SO These equations may be expressed as x v = Kx + O.y y, = O.x + K,y, or, expressing tbem in matrix form, r^"] \K 01 pc" LyJ 10 KJ Ll/J Therefore, tho matrix describing the transformation of en- largement is •R 0" K. 4. Botation through an angle 0. Let the point P move to the point P„ so that the vector OP t rotates through an angle 0. Let OPq = OP t ■= r (Fig. 79). Then r cos (* + 0) = r oos a cos 6 — r sin a sin 0. .*. as, = as oos 8 — y sin 0. and r sin (a + 0) = r sin a cos + r cos a sin 0. ,♦, y, = y cob 6 + x sin 0. This gives the following equations, as, = cos O.as — sin Q.y 3/, = sin O.as + cos B.y, Y P, Fio. 79 These may be written in matrix form as as. 'cos — sin 0' _sin OOS 0_ Thus, tho matrix describing the transformation is "cos —sin 0" sin cos 0, 81 MULTTPLICATIO?,- OF MATRICES Let A and B stand for two geometrical operations, and let AB represent the matrix which represents the result of per- forming the operations B and A, in turn. Let operation B send the point P to P v This may bo expressed P 1 = BP . Let the operation A send the point P 1 to P„. This may be expressed P 2 = AP t . If these two results are combined, it will be possible to write P t = ABP V ROTATION THROUGH 180° Method 1. Let P„ be the point {x, y) and let Pj be the point (*i> ^i)' separated from P by 90" (Fig. 80). Let P, be the point (iCj, i/j), separated from P„ by 180°. It will be seen that the following equations hold : 0, = -#, and y a = — y. Expressed in matrix form, these give (1) Thus the matrix describing a rotation of 180° is expressed i --1 0" -1. V JLi in matrix form as -1 -1 *L r t^./i\ X ^""^^ Ui.y 3 ) Via. 80 Method 2. P may have been transformed to P 2 by way of the point P,. 82 Thus: (a) Transform P„ to P, and obtain ro -r LsJ Ll 0. (b) Transform P x to P 2 and obtain ro -r LyJ (c) Now combine (a) and (b) and obtain La. 3- x% *0 -1" "0 -r ~x~ .Vt. .1 0. _1 o_ Jf. (2) By comparing (1) and (2) it follows that TO -1" o -n i o -i o 0" -1. Li oj This gives the result of multiplying together two matrices. The rule for the multiplication of matrices may be found in the following manner. Let x t = ax + jiy x s = ax l + by t Vi =yx + 6y y 1 = ex 1 + dy t . By eliminating x x and y^ from these equations, the following results will be obtained. x 2 = (a<* + by)x + (a/3 + bd)y y 2 = (c* + dy)x + (cfl + do)y. If these equations are written in matrix form, they become 'att + by ap + Sb d&"| 6d\ _ca + dy eft + This equation may be simplified to : A.B = C. The above gives a rule for the multiplication of two matrices. Example. Multiply To hi p q- ~P 9 and r » ~ap + br aq + be cp + dr cq + ds_ 83 The rule for the multiplication of matrices may be expressed hi words as follows. Let A be an I x m matrix (with I rows and m columns), and let B be an m x n matrix (with m rows and n columns). Then the product matrix, AB(=C) is an I x n matrix such that if C T , is the element in the r* row and «"■ column of C, it is the sum of the products of elements in the r" 1 row of A with the corresponding elements in the s" 1 column of B. This may be expressed symbolically as, m On = y Orlbi,. t-t It is necessary that the number of columns of A must equal the number of rows of B, or the product will not exist. ADDITION OF MATRICES The addition of matrices is defined for matrices of the same order. The sum of two matrices is the matrix which is obtained by adding together corresponding elements. Subtraction of ma- trices )3 defined by the subtraction of corresponding elements. Example 1. rs on ri 2i + = 7 8 3 4 5 + 1 7 + 3 C + 2" 8+4J 6 8' 10 12 Example 2. 1 6" 7 8 - "1 2~ 3 4 = "5-1 6-2" 7-3 8-4 "4 f 4 4 EXERCISE 12 1. Form the matrices A .B and B, A given that A = 2. Form the matrices A . B and B.A given that "1 T , B- "2 r Lo 3. £ 3. "2 -r , B = "3 2" Li 2. .2 ij 84 3. Multiply the following matrices 4. Find 5. If A = » r ir . o ij L o - r 2 ]_4 -3 (ii) o 6" ro oi 1 Lo ij 1 3' 4 2 3 2 -2' , find the matrices on the square which (i) .4 + B, {ii} 2A, (iii) A - B. 6. Use matrices to solve 3a; + y = 6, ox + 2y = 1 1 'i r 7. Find the effect of the matrix .1 1. lias its vertices at (0, 0), (1, 0), (1,1), (0, 1). 8. Given that % = ar s + y«, x« = x l + 2^, x % = 2ar — y a and y» = x t - y s , y,, = 2a;, - ft, y t = x + 2y a , express x t and y„ in terms of a-„ and y a . ANSWERS TO EXERCISE 12 "10 7 ■2 r "l 3" "8 1" 1. t 2_ , .12 9. L4 17. .' K _5 0. "l e " ~a b" "1 8" 3. (i) (ii) 4. _0 -1 . _c d_ .' ~1. ■a r 2 6" "-3 5" S. (i) i (") , (iii) • _4 7. 8 4. i ;. G. x = 1, y = 3. 7, The square is transformed into the line segment OF, whero P is the point (2, 2). J/3- - ■7 -r _1 7. J/p. or x 2 = 7.t — y and 3/3 = a- + 7#o. 85 CHAPTER FOETETEEH VECTORS A vector quantity is one which requires both a magnitude and a direction in order to specify it completely. A force and a velocity are examples of vector quantities as both a magnitude and a direction are needed to specify them. As force, and velocity, may be represented in magnitude and direction by a straight line, a vector could be defined as a straight lino dis- placement of a point. Vectors are denoted by smalt letters a, b, c, etc. in bold typo or by a pair of letters with an arrow above them, such as AB, CD, etc. Whereas a vector a is printed in bold type, the scalar quantity a is printed in ordinary type. A scalar quantity possesses magnitude only. The magnitude of the vector a is denoted by | a |, which is called the modulus of a. The modulus of a unit vector is unity, and the modulus of a null vector is zero. If two vectors a and b are equal, their moduli are equal and they act in the same direction. The vector which is equal and opposite to a is written as —a. The vector equal and opposite to AB is HA. (a) Consider the parallelogram ABCD, Fig. 81. — > — > Let AB and BC represent the vectors a and b respectively. Then AB ^ DC = a, and BC = AD=h, o Fig. 61 If AB represents the displacement of a point from A to B, and BC represents the displacement of a point from B to C, the 86 total effect of the two displacements will be the same as the displacement AC. That is AB + BC - AC, or, a + b = c . . . . (1) This gives the rule for vector addition. Since, also AD + DC = AC, then, b -i- a = c . . . . (2) From (1) and (2) a + b = b + a. This means that the commutative Jaw holds for vector addition. (b) Consider Fig. S2. Let AD, DC, and CB represent the vectors a, b, and c. o , c Fig. 82 / I Let a point be subjected to the successive displacements A D and DB. These will be equivalent to the displacement AB .-. AB - AD + DB. — >■ But displacement DB is equivalent to displacement DC + CB. .', AB m AD + {DC + CB). :. AB = a + (b + c) . . . . (l) Similarly, AB = AC + CB. 87 But AC = AD + T)C. AB = (AD -f DC) + CB. .: AB = (a + b) + c . . . . (2) From (1) and (2) a + (b + c)=(a+b)+ c. Therefore the associative law for addition holds for vector addition. (c) If A is a scalar quantity, or a number, and a is a vector, Aa is denned as a vector parallel to the vector a hut of modulus ?. times as great, » KINDS OF VECTOR A force needs magnitude and direction to specify it, and also its line of action must be stated. Such a vector is called a sliding vector. The magnitude and direction of magnetic field strength is referred to a particular point. This may be represented by a vector known as a tied vector. There aro quantities which are completely specified by mag- nitude and direction alone, such as the moment of a couple. Such a vector is called a, free vector. A system is a vector space if it is an Abelian group with respect to addition, is subject to a soalar multiplication by elements from an associated field of soalars, and if the scalar multiplication obeys the following laws: 1. Distributive law: r.(a + b) = r,a + r.b. -*■-*■•*■ (/■ + s).x = r.x + s.z. -> -> 2. Mixed associative law: r.&.a) — (r.e).a. 3. Unity element: I. a = a. Fio. 33 Figure 83 shows the representation of a vector by means of an arrow. The straight lino through P and Q is called the line of action of the vector. P is known as the origin of the vector, and Q the terminus of the vector. The letter indicating the origin precedes the letter indicating the terminus, and an arrow is placed over the letters. There is a simpler method of denoting a vector. This, as stated earlier in the chapter, con- sists of a single symbol, a small letter printed in bold type. The letter a in bold could denote a vector. The magnitude of a vector is a scalar quantity, and it is never negative. The magni- tude of the vector FQ is denoted by | PQ \ , and the magnitude of the vector a is denoted by a or by | a [. Two vectors are equal if they have the same magnitudes and the same direc- tions. If two vectors a and b are equal, the fact is expressed by a = b. ADDITION OF VECTORS Vectors obey a certain law of addition, called the law of vector addition, and this is illustrated in Fig. 84. Fio, 84 Let a and b be two vectors, and let the origin and terminus of a be P and Q respectively. A vector equal to b is now con- structed with its origin at Q. Let its terminus be at the point R. Then the sum a + b is the vector PR, and this fact is written a -j- b - PR. THEOREMS ON VECTORS I. Vectors obey the commutative law of addition. Therefore, a + b ■■ b + a. Proof. Let a and b be two vectors, as in Fig, 85. Then, a + b = PR 89 (1) Fio. 85 Now construct a vector equal to b, with its origin at P. Its terminus will fall on S. Now construct a vector equal to a with its origin at 5. The terminus of this vector will fall on R, Therefore, b + a = P B . . . . (2) From (I) and (2) it follows that, a + b = b + a. 2. Vectors obey the associative law of addition. Therefore, {a + b) + c - a + (b + c). Proof. Construct a figure similar to Fig. 86, with vectors a, b, and C as consecutive sides. Fig. 86 It will be seen that, (a + b) +C—PB+C—PS It will also be seen that a -f- (b + c) = a + QS = PS Therefore from (1) and (2), the theorem ia proved. 90 (1) <2) MULTIPLICATION OF A VECTOR BY A SCALAR By definition, if A is a positive scalar and a is a vector, the quantity Aa is a vector with magnitude Aa and acting in the same direction as a. Also, if /. is a negative scalar, /.a is a vector with magnitude | A j a, and acting in the opposite direction to a, 3. The multiplication of a vector hy a scalar obeys the distri- butive laws, Therefore, (i) (A + /t)a. = /a + /jo. . . - (1) (ii) A{a + b) = /.a + Ab . . {2) Proof (i) If }. -f /i is positive, both sides of (i) represent a vector of magnitude {). +.fi)a, acting in the same direction as a. If A + n is negative, both sides of (i) represent a vector of magnitude | A + /t | a, acting in a direction opposite to a. Fig. 87 Proof (ii) (a) Let n bo positive and let the vectors a and b be represented as in Fig. 87, and let the vectors [i& and /<b be represented as in Fig. 88. Then and /*{* + b) = f*PR fi& + fib «= SU . (3) W aa ub Fio. 88 91 The two triangles PQB and STU arc similar, and corres- ponding sides aro proportional, tho proportionality constant being ft. Therefore ftPR m SU. Since PR and SU have the same directions, and since ft is positive, then ftPR = SU. Therefore, fi{a + b) = /ta + ftb. (b) Now let ft be negative. Then Fig. 88 is replaced by- Fig. 89. Again, and ^{a + b) = fiPR, fia. + ftb = SU. T M-o y Fio. 89 The triangles PQR and STU are still similar, but the propor- tionality constant is | ft |. Therefore | ft \ PR = SU. Since PR and SU are opposite in direction, and ft is negative, then ftPE = SV. Therefore, j»(a 4- b) = pa. + fib. THE SUBTRACTION OF VECTORS If a and b are two vectors, their difference a — b is denned by the relation a — b = a + (— b). — b is denned as the vector with the same magnitude as b but acting in the opposite direction. Figure 90 shows two vectors a and b, and their difference a — b. Fro. 90 SCALAR PRODUCT Consider two vectors a and b with magnitudes o and 6. Let 6 be the smallest non-negative angle between a and [i, as in Fig. 91. Fio. 91 Then, the scalar product of a and b is the scalar ab cos 0. It is sometimes denoted by a . b. Therefore, a.b = ab cos 0. The scalar product is sometimes called the dot product. THEQBEStS OK SCALAR PRODUCTS 1 . The scalar product is commutative. That is a.b = b.a. 2. The scalar product is distributive. That is a.(b + c) = a.b + a.c. THE VECTOR PRODUCT Consider two vectors a and b, and lot the smallest non- negative angle between them bo (Fig. 92). Then, tho vector product of a and b is a third vector c which is defined in terms of a and b by the following conditions : (a) c is perpendicular to both a and b 93 Fig. 02 (b) The direction of c is that of the thumb of the right hand when the fingers point in the sense of the rotation of from the direction of a to the direction of b (c) c = ab sin 9 These conditions will define C uniquely. The vector product of a and b is denoted by a x b. Therefore, c a :■: b. The vector product is sometimes called the cross product. Theorem. The area A of the parallelogram with vectors a and b forming adjacent sides is given by A = | a x b I. Proof. Consider the parallelogram OABC, Fig. 93, where OA Fig. 93 represents vector a and 00 represents vector b. p is the per- pendicular from the terminus of b on to the line of action of a. Then, A = up. But, p = b sin 0, Therefore, A = ah sin — j a x b | . Theorem. The vector product is distributive. That is, a x (b + c) = a x b + a x c. Theorem. The vector product is not commutative, because, a x b = — b x a. 94 - BXEHCISE 13 1. The sides AB, BO, OA of a triangle ABC are denoted by vectors a, b, c. Prove that a + b -j- c = 0. 2. ABC is a triangle, and D, E and F are the mid points of the sides. The medians intersect at 0. Prove that OD 4- OE + OF = OA + OB 4 OC. 3. -4BCi5 is a quadrilateral. P is the mid point of BD, and Q is the mid point of AC. Prove that AB ~AD ~GB +CD = 4QP. 4. In the parallelogram ABCD, M and A T lie on the diagonal BD so thai By = MD. Show by a vector method that ANOM is a parallelogram. 5. A ship steams due west at 12 ra.p.h. A man walks at 4 m.p.h. relative to the deck from the port to the starboard bow. Find the actual speed and direction of the man's motion. [12-65 m.p.h., ^71° 34TV]. 6. Triangle ABC is right angled at B, and M is the mid point of AC. Tnking the position vectors of A and O relative to B as a and c, express BM and AM in terms of a and c. Hence, show that AM = BM. 7. Two circles touch externally at P. A common tangent touches the circles at B and C. Prove by a vector method that angle BPG is a right angle. 95 CHAPTER FIFTEEN INEQUALITIES The symbol > is used to donoto 'is greater than', and the symbol < is used to denote 'is loss than'. >■ means 'is not greater than' and > means 'is greater than or equal to'. # means 'is not equal to'. Much care is needed in the use of these symbols, and they cannot be used in tho manner used for the sign of equality, except under conditions to be described. The following statements will be self evident. (a) If a ~> b and b > c, then a > c (b) If a <C b and 6 < c, then a < c (c) If a > b, then a + x > b -j- x (d) If a > 6 and c > d, then a ~ c > b + d (e) Ifa<J and c < d, then a -j- c < 6 + d. Cautions, (i) if a ~> b andc < d, no relationship can be deduced regarding the magnitudes of a + c and 6 + d. (ii) Never multiply, or divide, inequalities except by num- bers known to be positive. (f ) If a > 6, and if A is positive, then la > ?.b (g) If a, b, c and d are all positive, and if a > 6 and c > d, then ac > M. (h) If a > 6, and if 1 is negative, then la < lb. Thus multiplication by a negative number reverses tho sense of inequality. Thus, if both sides of an inequality are multiplied, or divided, by the same negative number the inequality sign must bo reversed. An inequality may bo multiplied, or divided, by a positive number, or have terms transposed from side to side just like an equation. The modulus of a real number is its numerical value. A modulus is therefore always positive. The 'modulus of x % is denoted by the symbol | x | . To denote that a is numerically groator than b, the modulus notation is used, and it is written \a\ > | b \ . The solution of inequalities. (1) Lot x ~ 2 > 7. (1) Subtract 2 from both sides: Then x > 5. 96 (2) Let 2a: > -S. Divide both sides by 2 : Then m > -t {3> Let f x < 4 Multiply both sides by 5 : Then a; < 20. (4) Let -ix < -f. Multiply both sides by 2 : Then -x < ™|. Multiply both sides by —1: this changes the sense of the inequality. Then x > J . PICTOKIAL REPBESENTATIOX OF DOv QUALITIES H 1 1 1 1 I h— 1 1 I \~ Fig. 94 -5 -4-3-2-1012345 Figure 94 represents part of the set of natural numbers on a number line. Points on this line to the right are greater than points on the line to the left. S A . -* - • • f Fie, 95 - 1 - H 1 1 1 1 h -I r- -5 -4-3-2-1012345 Figure 95 shows part of the number line representing the natural numbers. The lino A represents the solution set x > 3, and the lino B represents tho solution set m < — 2. Figure 96 shows a two-dimensional plane figure. The line --1 H h -~-2 Fia. 96 97 FQ represents the set of points x <m 2. The shaded region shows the set of points a; > 2. This is an infinite set. Figure 97 shows the line ItS which represents the set of points y = —2, and the shaded region shows the set of points y < — 2. This is also an infinite set. r --3 --2 --1 -1 1 h H \-x Fio. 97 ---1 Fio. Figure 98 shows the line TV which represents the set of points y = x. The region shaded horizontally shows the set of points, y < x, and the region shaded vertically shows the set of points y > x. m P is the point of intersection of AB and CD. The symbol for the intersection of two sets is C\. Therefore the intersection of the solution sots of y = as + 2 and y = 2x — 3 may be repre- sented by, {(x, y) | y - x -f 2} n {(as, y) \ y = 2s - 3}. In Fig. 100, the line EF is the graph of y = a; + 3. All points in the region shaded horizontally are members of the solution set y < x -+■ 3. This excludes the points actually on EF. Fig. 98 IN-TEBSECTKJG GRAPHS In Fig. 99 the line AB is the graph y = x + 2. All points on it aro members of the solution set of the equation y = x + 2, The lino CD represents the graph of y = 2a;— 3. All points on it are members of the solution set of the equation g — 2a; — 3. Fig. 100 hrH h* The line GH is the graph of y = —x -J- 1. All points in the region shaded vertically are mombora of tho solution set V < — x + 1. This excludes the points actually on OH. The intersection of tho solution sets y < x + 3, and y < —x + 1 is the set of points in the doubly shaded region. These may bo represented symbolically by, W*, y)\y <x + 3}f\ {{x, y) \ y < -x + 1}. EXERCISE 14 1. Find the values of x for which fa: + £ < 2x + £. 2. Find the values, of x for which a; 4 — 5x + 4 lies between -2 and +2. 3. Find the values of a: for which 14a; — 20 — 2x a is greater than 5. a; — 7 4. Find the values of a: for which 2 < < 3, x - 2 2 5. Find the ranges of a: for which 5 is greater than zero. x — 4 6. Find the range of x for which 3 -) is less than zero. x + 2 2a? 4- 3 7. Find the values of x for which — 1 < < 1. x — 1 ANSWERS TO EXERCISE 14 1. For values of x less than — 1. 2. |(5 - VYt") < as < 2, and 3 < x < f. (5 + vT7> 3. None. 4. -3 <a: < -f. 5. a; < 4 or a; > 4f . 6. -2 > x > - 3f . 7. -4 <k < - f. 100 CHAPTER SIXTEEN FINITE ARITHMETIC Consider the natural number 3. There are two families of numbers which are not multiples of 3. One family consists of numbers which are 1 more than an integral multiple of 3. For example 7 and 10. The other family consists of numbers which are 2 more than an integral multiple of 3, for example 11 and 17. All integers, therefore, will fall into one of three classes, depending upon whether the remainder, when the integers are divided by 3, is or 1 or 2. These three classes are called residue classes modulo 3. The three classes may be listed as follows : (a) Class 0. -12, -9, -6, -3, 0, 3, 6, 9, 12, 15 . . . (b) Class 1. -11, -8, -5, -2, 1, 4, 7, 10, 13, 16 . . . (c) Class 2. -10, -7, -4, -1, 2, 5, 8, 11, 14, 17 . . . The class has properties not possessed by the other classes. Thus, the sum of any two members of the class is also a member of that class. This is tho closure property, which means that addition is a binary operation for the class 0. It will also be seen that the class contains the identity element for addition, which is 0. Furthermore, for every element in the class its negative also belongs to the class. All this means that the class satisfies all the requirements for being a group. Since addition is commutative it is also an Abelian group. Thero is another interesting property of tho class 0. If any two mombers of tho class are multiplied, the product is also a member of the class 0. Thus, multiplication is a binary opera- tion for class 0. As it obeys the distributive law, tho class satisfies the requirements for being a ring. If any member of class is multiplied by any integer, whe- ther in the class or not, the product is a member of class 0. A sub ring which has this property is said to be ideal. However, the class does not contain the number 1 which is the unity element for multiplication. Thus a ring need not have a unity element. 101 RESIDUE CLASSES MODULO 3 In order to construct a now class, it is necessary to define addition and multiplication. 1. Definition of Addition. To add two residue classes, choose one member from each class, and add these members. The class to which the sum belongs is called the sum of tho residue classes. 2. Definition of Multiplication. To find tho product of two residue classes, multiply any member of one class by any member of the other class. Tho class to which the product belongs is the product of the classes. It is now possible to construct two tables for class 0, ono for addition, and one for multiplication. Addition Table Multiplication Table X 1 2 1 1 2 2 2 1 SUMMARY OF THE PROPERTIES OF RESIDUE CLASSES MODULO 3 1. A system of residue classes obeys the following five Jawa. (a) It contains an clement called 1. (b) For every member in the system, there is another member, and only one, called its successor. (c) Two distinct members do not have the same successor. (d) There is no member of the system which has 1 as its successor. (e) If a set of elements belonging to tho system contains 1, and, for each member that it contains, also contains its successor, then this set contains the whole system. Therefore, a number system is made up, consisting of only three elements, which is both a group and a ring. 2. It is a group with respect to addition because: (a) It contains a zero element, i.e. the class 0, 102 (b) For every element in the system, thore is also a negative in the system. Tho class is a zero element because that class added to any other class leaves it unchanged. The group iB Abelian because the addition operation is commutative. The system is also a ring because it has a multiplication operation which is distributive with respect to addition. RESIDUE CLASSES MODULO 6 When an integer is divided by 6, the remainder is either 0, or 1, or 2, or 3, or 4, or 5. Therefore there are six residue classes modulo 8. The remainder associated with each class is the name of the class. If addition and multiplication are defined as they were in the case of residue classes modulo 3, tho following tables may be constructed: Addition Table Multiplication Table + 1 2 3 4 5 1 2 3 4 5 i ; i ! 2 ; 3 1 1 4 5 i|i 3 4 5 ] 3 , 3 4 5 1 2 4 4 5 1 2 3 '550 1 2 3 4 X 1 2 3 4 5 1 1 2 3 4 5 2 2 4 2 4 3 3 3 3 4 4 2 4 2 5 5 ' 4 3 2 1 RESIDUE CLASSES MODULO 5 The integers modulo 5 are : 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, . . . These are obtained by dividing the natural numbers by 5 and writing down the remainders. ITU — H *™ It will be seen that these additions hold : 2+3 = 24-4=1 3+4=2 4+4=3 The following addition table may be constructed : + 1 2 3 4 1 2 3 4 1 1 2 3 4 2 2 3 4 1 3 3 4 1 2 4 4 1 2 3 ±ho properties are as follows: 1. Closure. The sum of any two elements is also an element. 2. Identity Element. An element which does not alter the ele- ment upon which it operates is 0. 3. Inverse Element. For each element there is one element, and one only, which cancels its effect. Take 0, and add another number, say 3. It is possible to find another number, which is 2, which, on addition, brings the value back to zero; and 2 is the only number which will do this. 4. Associative Law. For any three numbers a, b, c, it is possible to write a + (6 + c) = (a + b) + c. Properties 1 to 4 are the properties of a group. So the inte- gers modulo 5 form a group. It is possible to show that the integers {0, 1, 2, 3, 4} modulo 5 do not form a group under multiplication, but the integers {1, 2, 3, 4} modulo 5 do form a group under multiplication. exercise 15 State whether, or not, the following addition r 1. 2. 3. 4. 5. sots are groups under The integers modulo 8. The integers modulo 6. The integers modulo 3. Tho real numbers. Tho negative numbers. State whether, or not, the following sets are groups under multiplication, in each case exclude 0. 6. The integers modulo 8. 7. Tho integers modulo 7. 8. The integers modulo 6. ANSWERS TO EXERCISE 15 1. Yes. 2. Yea. 3. Yes. 5. No. 6. No. 7. Yes. 4. Yes. 8. No. 104 105 CHAPTER SEVENTEEN SYMMETRY GROUPS For many objects to be pleasing to the eye, they must possess symmetry and the right proportions. An oblong picture must be such that the length bears a definite ratio to the width for the effect to be most pleasing. The bulge in a vase must bo greatest at a certain fraction of the height. Such facts wore well known thousands of years ago. Quito primitive people constructed patterns with simple shapes such us circles, triangles, and squares. The Romans uaed pattern and symmetry in their tessellated floors and in their buildings. Symmetry is closely related to reflection and rotation; it can be obtained by the reflection of a figure in a mirror, so that both sides of the line of symmetry are identical. Some kinds of symmetry are produced by rotation about a point. Many letters of the alphabet possess symmetry. i i i i ill! The letters A, H, M, T, 17, V, W, X, Y are obtained by re- flection in a vertical mirror. -B0-EK- The letters B, D, B, K are obtained by reflection in a hori- zontal mirror. NSZ The letters N, S, Z are reproduced if they are rotated through an angle of 1 80°. SYMMETRY OPERATIONS Let p denote a half turn about a horizontal axis, or a rota- tion through 180°, and let q denote a half turn about a vertical axis. 106 The operation p, and the operation q, on the letter Z both produced the letter ^ . Therefore the operation p followed by the operation q will reproduce the original Z. However, Z may bo mapped on to itself by a single oporat ion, a rotation of ISO 3 , which is a half turn, in the plane of the paper. Let this latter operation be denoted by the letter r. It is now possible to describe the motion symbolically : p[Z] -> X- Also, q . p[Z] — >- Z. However, r\Z\ — >■ Z. Therefore, q.p = r. It may be shown, in the same manner, that p.q ~ r, and also thatp.p, orp ! , = t. THE IDENTITY ELEMENT AND THE INVERSE ELEMENT There are letters of the alphabet, such as F, G, J, L, P, H, which are unsyrametrical. These may be brought into line by considering another operation denoted by J, known as the identity element. Operation with I leaves the letter unchanged. Since operating by p twice, orp s , restores to the original, it will be seen that p* = I. If an operation s is reversed by a further operation t, it is said that the operation i is the inverse of the operation s. THE GROUP OF SYMMETRIES OF THE RECTANGLE (i) Let p represent a half f urn about a horizontal axis, (ii) Let q represent a half turn about a vortical axis, (iii) Lot r represent a half turn in tho plane of the paper. A D B C P c A B _£^ C 3 D A y\ X A D B C <? A C D P t C P D A Fio. 101 Figure 101 shows the operations by means of which one position may be changed into another. 107 If a rectangle is out from paper and lettered, the following rules may be verified : p.q = q.p = r. q.r = r.q = p. Also, it is evident that, J.p=p; I.q=q; I.r=r; i 3 = I. The results may bo placed in a 'multiplication* table p.r = r.p = q. p* =q* =r* = I. X 1 P q r First operation I I P n T Second p P I T 1 operation 9 ? T I P T r 1 P I It will be aeon that the symmetries of the rectangle have the following properties: 1. The set has a unique identity element I. This means that there is only one element, I, which, when multiplied by, or multiplying, any element of the set leaves that element unaltered. I.p = p.I =p; I.q =q.I = q; I.r = r.I = r; I.I = /. 2. The set possesses the 'closure' property, since the product of any two elements is also an element of the set. 3. Each element of the set has a unique inverse element. Actually each element is its own inverse element. The effect of the operation p is cancollod by a second operation p. p.p =p* =1; q.q = q* = I; r.r = r*=I; I.I =!*=!. 4. The associative law is obeyed. Example. p x {q x r) ■= p x p = J, (p x q) X r m r x r = J. .-. p x(q xr) =(p xq) X r, and so on for any three elements. 108 The four properties listed above show that the symmetries of the rectangle form a group. 5. There is an additional proporty, the commutative law, which is also obeyed, since: p.q <= q.p; p.r = r.p; q.r=r.q. Such a group is called a commutative group, or an Abelian group. SYMMETRIES OF THE EQUII^TERAL TRIANGLE This is a good example of a finite group. The symmetries of the equilateral triangle are motions which bring it into coinci- dence with itself. The best way to study the motions is to begin by cutting out of paper an equilateral triangle, and labelling its vertices A, B, and C, as in Fig. 102, Mark the letters on both sides of the triangle. Place the triangle as shown, with EC hori- zontal and below A. Each motion which brings the triangle into coincidence with itself will be given a name. Fio. 102 (a) One motion will be a clockwise rotation of 120° about the centre of the triangle. This will bring B to A, A to C, and OtoB. Call this motion P. (b) Another motion will be a rotation of 240° in a clockwise direction. This will bring C to A, B to O, and A to B. Call this motion Q. (c) Another 'motion* will be a rotation of 0°, involving no movement at all. Call this 'motion* /. There are three fur- ther motions which will bring the triangle into coincidence with itself. (d) Raise BO and turn it over the line AC, which remains where it is, so that the bottom face comes uppermost. See Fig. 103. 109 Fia. 103 The motion keeps A where it is and changes the positions of B and C. Call this .motion R. (e) Figure 104 shows a similar motion about B, in which A and C change positions. Call this motion S. Fig. 104 (f) Figure 105 shows yet another similar motion, about C, in which A and B change positions. Call this motion T. Fig. 105 Tho foregoing has built up a system of six elements, F, Q, I, R, S, T. Now define a binary operation © for the system thus: If X and Y represent any two of these motions, the product 110 X © Y is the motion which results when the two motions are performed one after the other, Y being performed first. The result of performing the operation © on the equilateral triangle is listed in the following table. The motion which is performed first, and written on the right-hand side of a prochict, is indicated at the top of the table, and the motion which is performed second, and written on the left-hand side of tho product, is indicated down tho left-hand Bide of the table. © I P Q R S T I I P Q ' R S T p P Q I S T R Q Q I P T R S R R T S I Q P S S R T P I Q T T S R Q P I In order to show that the symmetries of the triangle form a group it is necessary to show that: (a) The operation © is associative, i.e. that X © (Y © Z) = (X © Y) © Z. (b) The system has an identity element; this is I". {c) For every element in the system there is an inverse element also in the system. The inverse of P is Q, and the inverse of Q is P. since P®Q=Q®P=I. As these are all verifiable from tho table, then the symmetries of the triangle do form a group. ISOMORPHIC GROUPS Comparing the characteristic groups of symmetries which two different sets of elements possess it will often be observed that there is a similarity of structure. Tho comparison may be made using the addition and multiplication tables. When two groups have tho same basic structure, they arc said to be isomorphic. Ill EXERCISE 16 Draw the letter G after the following operations have been performed upon it : 1. p 2. q 3. r i. p* 5. 9* 6. f a 7. /. 8. p.q 9. q.p. 10 The operation m means 'rotate through 1 20' anticlockwise about the centre of gravity in its own plane'. Perform tho operations J, a), and w* on the equilateral triangle * Then complete the multiplication table e^—^c X 1 W £j' 1 <u Ol* 1 1. Given that p represents a half turn about the perpendicu- lar from A to BC, q represents a half turn about the perpen- dicular from B to AC, and r represents a half turn about tho perpendicular from C to AB, perform the following operations on the equilateral triangle ABC. >a . p or. C L- \ 3 (o> q art g A ft (dj I or. B L\ A ANSWERS TO EXERCISE 16 A 10. B C A, Ac A 11. ia.y B l\ c (W a L-\ A' — *fl C a A a co A A c 112 x I [ U »' * 1 1 1 u u* W U '.»-* 1 a>! u' < H (::> A CHAPTER EIGHTEEN SENTENCE LOGIC PROPOSITIONS In this ohapter letters are used to represent statements or propositions. Thus, the statement T have had breakfast' could be represented by the letter a, and tho proposition 'I have had dinner could be represented by tho letter 6. If it is midday it is likely that a is a true and b is a false proposition. In that case we write; a = 1; b = 0. Thus, the symbols 1 and are here used to denote the truth or falsity, respectively, of propositions. Just as in the case of switching algebra, there are no half truths. Thus, a sentence is not a proposition unless it permits of a definite decision of its truth or falsity. COMPOUND PROPOSITIONS The statement 'I have had breakfast and dinner' is a com- pound proposition containing two simple propositions, both of which are true. If either meal or both meals wore missed, the compound proposition is false. If the letter c is used to represent the compound proposition a truth table can be constructed as follows: a b e 1 1 o o 1 1 1 This table is the same as the multiplication table for the numbers 1 and 0. It is also the same as the table for a circuit with two switches in series. Thus it is possible to write : ab = c. This combination of propositions is named 'conjunction'. It states 'both a and 6'. Another combination, called 'disjunction*, states 'either a or 6 or both*. 113 If a man comes home and is asked 'have you had breakfast or dinner', and ho replies 'ycs'> he is stating that lie has had one or both of those meats. The compound proposition, say d, would be false if, and only if, both a and b are false. The truth table would then be: a ft d 1 1 1 1 I 1 1 This table is the same as that for a circuit with two switches in parallel. Thus it is possible to write: a -t- 6 = d. NEGATION OF A PROPOSITION The negation of a will be the proposition 'I have not had breakfast', and the symbol used is a'. It will be obvious that if a = 0, then a' = 1, and if a — 1, thon a' = 0. If a man is asked 'have you. had breakfast or dinner', and answers 'no', then he is stating the proposition (a + ft)'. This is the same as the conjunction of the two propositions (i) I have nob had breakfast, and (li) I have not had dinner, which may bo written as u'b'. Therefore, (a + 6)' = a'b'. IMPLICATION The symbol => means 'implies* or 'if ... . then'. Thus p ;*■ q means 'if p, then q\ Suppose wo say 'if you get wet you will catch cold'. Here p = (you get wet) and q = (you will catch cold). Let the compound expression p => q be denoted by r. If, however, one gets wet and does not catch cold, the statement is false, then ** = 0. So two rows of the truth can be written : P q r 1 1 1 1 114 If one does not get wet, and does not catch cold, then: p q r 1 However, one could get a cold without getting wet, so p 9 r 1 1 Thus, the complete truth table will be : STATEMENTS ItEPKESENTED SYMBOLICALLY Consider two statements represented by p and q. (i) The negation of p is written <—>p (that is, not p). For example, if p stands for 'the sun is shining', then ~p stands for 'the sun is not shining', (ii) p /\q represents p and q. For example, if p stands for 'Jack likes Latin', and q stands for 'Jack likes French', then p A 2 stands for 'Jack likes Latin and French'. There is an analogy between sets and statements. The laws by which statements are manipulated are the laws of sets. That is why there is a similar form of notation. Thus, if P is the set of all children who like Latin, and Q is the set of all children who like French, then P Pi Q is the Bet of all children who like Latin and French, (iii) fV? stands for p or q or both. For example, if p stands for 'Jack is shouting' and q stands for 'Jack is clapping', then p V <? stands for 'Jack is shouting or clapping or shouting and clapping'. 115 (iv) p ->q moans the statement p implies tho statement g. For example, if p stands for 'Smith is a Londoner* and g stands for 'Smith is an Englishman', then p ->• g, because the first statement implies the second. However, we cannot write q -*-p, because not all Englishmen are Londoners, (v) If wo have two statements 30 and q which are such that p implies q and q implies p, then the statements are said to be equivalent, and this is written p «-»■ 7, For example, if p stands for 'triangle XYZ is equilateral', and q stands for 'triangle XYZ is equiangular*, then p implies q, and q implies p, and we may write p «-► q. TRUTH TABLES TIi ere are occasions when it is not easy to decide whether or not a sentence 13 true. It is then useful to construct a truth tabic. (a) If a statement p is true {2'), then tho statement ^p must be false (F), and vice versa. From this a simple truth table can be constructed. P ~P T F F T (b) Consider p A ?■ There are four possibilities: (i) p and q are both true (ii) p and q are both false fiii) p is true and q is false (iv) p is false and 7 is true. Since 'p and g' is true only if p and q are both true, the fol- lowing truth table can be constructed : p 1 p A? T T T T F F F T F F F F 116 (c) Consider p\J q.lsx this case either p or q or both are true. If ji is false and g is true, or vice versa, the statement, or dis- junction, is true. For example: (i) 'Either men have two legs or dogs have two legs'. This is a true statement because one of the facts is true. If both statements wero false, then tho whole sentence would be false. (ii) 'All men are black or all birds are white' is obviously a false statement. A complication arises when both statements are true. If a man is wearing a green coat and a brown hat, would it bo true to say 'he is wearing a green coat or a brown hat'? Xow p V 9 has been defined as p or g or both, which is called the inclusivo disjunction. It is possible to say 'he is wearing a green coat or a brown hat or both'. If both statements are true, the compound sentence is true. If however, p means 'I am in Australia', and q means 'I am in Scotland', then the inclusive disjunction is meaningless, and for such cases the symbol V w used> which means p or q but not both. Now it is possible to construct the following truth table: p 9 pVi T 2- T T ¥ T F ¥ T F F F (d) Tho symbol -»■ means 'if ... . then', or 'implies'. So that p ->■ q means 'if p then g* or 'p implies g', If p and q are both true, then p ->■ g is true. If p is true and q is false, then p -*-q must be false. It is not obvious what happens when p is false, so we define that whenever p is false then g is true, and the follow- ing truth table is constructed : P 1 P-+1 fft rT* J7 1 T F F F T T F F T 117 (e) The symbol p*-*q means 'if p then q and if q then p\ p and q can be equivalent statements only when both are true or when both are false. Thus the following truth table can be constructed : p ? P«-»ff T T T T F F F T F F F T ILLOGICAL SEASONING There is much confusion between a statement and its con- verse, i.e. between p =-17, and q => p. The statement 'If ho is ill, then the doctor will call on him', has as its converse: 'If tho doctor calls on him, then he is ill'. If tho first statement is accepted as true, it does not neces- sarily follow thut the second statement is true. The proposition p' => q' is said to be the inverse of p =*■ q, but this is not a logical sequence. For example, consider the statement 'If he is an African, he is black'. The inverse is, 'If he is not an Afriean, he is not black'. The inverse is not true. However, a s> 6, and b' =- a', have the same truth tables, and so the statements arc equivalent. EQUIVALENCE If one wishes to assert both a and b, tho symbol p <& q is used, poq means that p and q are either both true or both false. Then p and q are said to bo equivalent. The truth table for p <?■ q is : • ■ pOq 1 c 1 1 1 1 1 Example. Suppose p = triangle ABC has two equal sides, and q = triangle ABC has two equal angles, then it follows p -o- 5. If one proposition is true, so is the other. If one proposition is false, so is the other. Example. Let p = triangles PQR and XYZ are congruent, and let q = triangles PQR and XYZ are equal in area. It is known from geometry that p ^ q, or, in words, 'p is a sufficient condition for q'. This gentencti must not bo oonfusc-d with tho similar one "q is a necessary condition for p'. This sentence means exactly the same as the former. It is anothor way of describing p^r q, Tho triangles must be equal in area for them to be congruent. But the equality of area is not sufficient to be sure of congru- ence, and so 'q is not a sufficient condition for p'. Also, it is not necessary for two triangles to be congruent for them to be equal in area, and so 'p is not a necessary condition for q\ It can happen that a condition is both necessary and aufli- oiint . In order that a triangle be isosceles, it is a necessity that two angles should be equal, and this is a sufficient condition. The symbol -o describes the situation. This may also be des- cribed by using tho phrase 'if and only if. Then, one says, a triangle is an isosceles triangle if and only if two angles are equal. BOOLEAN ALGEBRA, SETS, AND LOOIO Comparison of the symbols used in sets, sentence logic and switching circuits. Union either -or or both Intersection and Sots A\JB Sentence logic eV ' Switching circuits a + b AC\B a A 6 0.6 Complement negation a,' 118 The algebra of sentence logic and the algebra of switching circuits both obey the laws of sets. They are all examples of Boolean algebra. V is the universal set. A, B, and C are subsets in Boolean algebra. 1 is written instead of V , and is written instead of 0. mi-i 11" Laws of Boolean Algebra a x 1 = o o + 0=a o(6 + c) = o6 + oe a -~ be - (o + &)(« + c) a(a + 6) = a a + at = o ox0 = o+l=l a x a = a + 0=0 {o + 6)' = a'.b' {ab)' = a' + b'. Laws of Sets 1. AnU =A 2.iU0=l 3. A f\ (B V C) = (A n B) U (4 A C) 4. iU(5n C) = (iU£)n(AUC) 5. A H (4 U B) = A 6. A U {A r\ B) = A 7. A n = 8. 4 U C = U 9. Ar\A =A 10. AvA =A 11. (AUB)' =A'C\B' 12. U n B)' = A' U B' The principle of duality is an important one in mathematics, and it is well illustrated by the above laws. If any rule is taken and + and x are interchanged and and 1 are interchanged, another rule which is true is obtained. Observation wilt show that the following rules are dual, (i) 1 and 2, (ii) 3 and 4, (iii) 5 and 6, (iv) 7 and 8, (v) 9 and 10, (vi) 11 and 12. Thus there are six pairs of self dual relationships. EXERCISE 17 1. p represents the statement *I like sailing' and q represents the statement 'I am a good swimmer'. Write the meaning of the following: (!) P A § t"') P V 9 (iii) V A "-' 1 (iv) ~p V q (v) ~P A ~3 (vi) p ->?. 2. a represents the statement 'I am wearing a brown coat' and b represents the statement 'I am wearing brown shoes'. Write the meaning of the following: (i) f" a {ii) o V "•' * (' u ) b /\ ~b (iv) a V ~ ° (v)oA~i (vi) a-*- b. ANSWERS TO EXERCISE 17 1. (i) I like sailing and I am a good swimmer. (ii) I like sailing or I am a good swimmer (or both). 120 (iii) I like sailing and I am not a good swimmer. (iv) I do not liko sailing or I am a good swimmer {or both), (v) I do not like sailing and I am not a good swimmer, (vi) I like sailing implies I am a good swimmer, (i) I am not wearing a brown coat, (ii) I am wearing a brown coat or I am not wearing brown shoes, (iii) I am wearing brown shoes or I am not wearing brown shoes, (iv) I am wearing a brown coat or I am not wearing brown shoes, (v) I am wearing a brown coat and I am not wearing brown shoes. (vi) I am wearing a brown coat implies I am wearing brown shoes. 121 CHAPTER NINETEEN POINTS, RELATIONS AND FUNCTIONS The set of values of x for which x is greater than three is written symbolically as {x \ x > 3}. If the set <f is (1, 2, 3, 4, 5, 6), then for x e S, {x | x > 3} is the set (4, 5, 6). If x may take values which are positive whole numbers between and 10, that is x e «f, where S = (1, 2, 3, 4, 5, 6, 7, 8, 9), then the set {x | x > 3} is the set {4, 5, 6, 7, 8, 9}, If however 3 is the set {0, 1, 2}, then {x\ x > 3} = 0, which is the null set. Example. Assuming x may take only positive integral values, list the set {x \ x > 2} U \x \ x < 3}. {x | x > 2} U {x | x < 3} = {3, 4, 5, 6 . . .} U {1, 2} = {1,2,3,4,5,6...}. ORDERED PAIRS Let x and y be typical elements of two sets X and Y, so that x e X, and y e Y. The set (:*:, j/) is an ordered pair, and is 80 called because of the importance of the order in which the elements are written. For example, the points (3, 4) and (4, 3) are not the same. This is because, by convention, the as-co- ordinate of a point is placed before its {/-coordinate. RELATIONS A set of ordered pairs is called a relation, becauso there is usually a definite connection between the elements of each pair and applying to all pairs. To draw the graph of a relation, two scales are needed, one for the first elements and one for the second. For this purpose two lines are drawn at right angles, and they are called axes. The set of first elements ia marked along the horizontal axis, and the sot of second elements is marked along the vertical axis. The ordered pairs are thon represented by points on the diagram. The domain of a relation is defined as the set of all the first elements in the relation, while the set of all the second elements in the relation, is called the range of that relation. If j) represents any first element of a relation, nnd q roprosonts a corresponding second element, then the ordered pair (p, q) represents a typical member of the relation. The relation men- tioned here is a binary relation because it relates two things. 122 It is possible however to have relations connecting more than two things. A first element may have many second elements associated ■with it. It is then called a many-to-ono relation. A second element can have only one first element associated with it. LINES AND REGIONS Consider tho sot of ordered pairs {[w, y) [ x -f- y — 3} in which there is no restriction on the values of x and y. The set will contain an infinite number of members, each of which will lie on tho straight line AB (Fig. 106), which is called the graph of the equation x + y = 3. (TV v 4 £ 1 - 1 2 ^S 4 Fig. 108 Fig. 107 If a; and y are real numbers, tho sot of ordered pairs {{x, y) | x + y > 3} will fill the whole plane shown shaded in Fig. 107. This region will not contain the boundary line x + y = g. The boundary will only be included if the set wore {{x, y)\x+y > 3}. FUNCTIONS A relation in which no two different ordered pairs have the same first member is known as a function. Thus {1, 1), (2, 3)} is a function, but not so {(1, 2), (1,3), (2, 3)}. A function j:A~-> Xi, from a set A into a set B, is a correspondence that assigns to oaeh element a e A & unique element in B denoted by /(a). Tho symbol /(a) is called tho valuo of/ at a. The range of a function f:A -> Ti is a subset of the co-domain B com- prising elements that actually correspond to some element of A, 123 Example, {(x, y) \ y = x + 8} b a function because if a list of ordered pairs b drawn up, there are no two pairs which have the same first but different second element. To each value of y there b one and only one value of x. In thb particular case, there b a correspondence between the set of real number values of a; and the set of real values of y. Example, {(x, y)\y > x -f 2} b not a function because if a list were made of ordered pairs, it would contain such as (1, 3) and (1, 4), etc. Example. {{x,y) \ x = 10} is not a function because a Ibt of ordered pairs would contain such as (10, 1), (10, 2), etc. Example, {{x, y) j y = 7)} is a function, because no different ordered pairs of the set (1, 7), (2, 7), (3, 7), (4, 7) . . . contain the same first element. Example, {[x, y) | y* = x} b not a function because a lbt of ordered pairs would contain such as (4, —2) and (4, +2). However, if the parts of the curvo y % — x in the two quadrants are treated separately so that we have CO {(*.*/) or(ii) {[x,y) y* = *, y > 0} y = +Vx), then each of (i) and (ii) forms a function. COtt HENPOSBEKCE Correspondence between sets has considerable importance in mathematics. That type of correspondence in which every ele- ment in a domain has exactly one partner in the co-domain is called a function. 6 • M A B s (b) l^l tc) Fig. 108 m Consider the correspondences in Fig. 108. In Fig. 108(a) the element 6 of A has no corresponding element in B. Therefore the matching b not a function from A to B. In Fig. 108(b) each element of A does have a matching value in B. Abo, no element of A has more than one corresponding value in B. Therefore the matching is a function from A to B. 124 In Fig. 1 08(c) every clement in A has a corresponding element in B, but to the element a in A both p and a are matched. Therefore the correspondence b not a function from A to B. In Fig. 108(d) every element of A has exaotly one correspond- ing element in B, no more and no less. Therefore the corres- pondence is a function from A to B. In the above, the set A is called the domain of the function, and the set B is called the co-domain. A function b a pairing of the elements of two sets in which every element of the domain b used exactly once. Every element of the domain must be used, and used only once. INVERSE CORRESPONDENCE If there b a function from .4 to B, there b a correspondence or matching from A to B. If there b a correspondence from B to A, that b in the opposite direction, it b known as the inverse correspondence. Considering the examples in Fig. 108, it will be seen that the inverse correspondence ta not necessarily a function. The symbol for the inverse correspondence is /~ l . /(a) is sometimes called the image of o under/, and/ -l (6) is called the inverse image, when b b the corresponding image in the co-domain. MAPPING A mapping b a function. Just as the points or elements on a map correspond to points or elements of a piece of land. Every field or every building will be represented by one and only one part of the map. Lot the elements corresponding to buildings be denoted by B, and let corresponding parts of the map be denoted by M. Then, the mapping can be symbolized by f:B -* M. If be B and m e M, then it is possible to write f(b) = m. In this particular mapping there is one: one correspondence. Also, m b the image of b in the set M. exercise 18 State whether the following sets of ordered pairs are relations or functions: 1. {fr l y)\x + y = 5} 3. {{x,y)\y-5=0} 2. 4. 125 {(«. v) x +y > 5} x + 3 = 0} {(*» y) y-x*-0} 6. {{x,y)\xy^=9} a;" + y* = 36} 8. {(», y) j a: 1 + y* < 36. In each of the following functions, state whether or not the mapping is 1 : 1 and state the domain and range : 9. {{x, y) | x + y = 6} 10, {<*, y) H. {(x, y) | y * - x = 0} 12. {(«, y) 13. {(», V ) j xy - 9} 14. {(ar, y) j/ - 6 = 0} y = x*} y =x* + 1}. ANSWERS TO EXERCISE 18 1. A function 2. Not a function 3. A function 4. Not a function 5. A function 6. A function 7. Not a function 8. Not a function In the following, z is the set of real numbers and z+ is the set of positive real numbers; 9. 1: 1; z; z 10. Not I: 1; z; 4 11. 1:1; z; z 12. Not 1:1; z; z+ 13. 1:1; 14. Not 1 : 1; z; all real numbers > 1. T26 CHAPTER TWENTY STATISTICS Statistics is the study of the collection and meaning of data. Number facts are called data. The complicated modern way of lifo often involves statistics; used by scientists to solve prob- lems, by politicians to 'p rove ' arguments, and by commerce to obtain information about business. They deal with sport, the weather, prices, wages, and numerous other things. They are used to influenco our thoughts and our lives. The arguments that smoking causes lung cancer are entirely statistical. The number of people who vote for a particular party, or use a particular brand of article, or watch a particular television pro- gramme, is deduced for the whole country by taking a sample; by putting questions to a certain number of people, and then using proportion. It is important to uso discretion in 'sampling' because 'bias' may be introduced. Statistics are often misinter- preted deliberately, to produce a desired result. Sometimes they are misinterpreted because certain factors are not taken into account. All statistics must bo viewed with great car© and the true meaning can only bo obtained by asking such questions as; who collected the date, what was the source of the data, what was the method of collecting the data, and so on. School reports or marks obtained in an examination, can be particu- larly misleading. A student who obtains 60% in mathematics could be top of the class or bottom of the class. The top boy in form 4C may be weaker than the bottom boy in form 4A. Yet the former boy may have 'good' on his report, and the latter may have 'weak' on his report. The marks obtained in an examination taken by many thousands may be most mis- leading. A 'pass' could range between 50% to 100%, and a 'fail' be anything from 0% to 49%. Some subjects are easier to pass than others. Also we have 'off* days. One can get a reasonably true picture of the performance of a fit pupil only by considering how many pupils obtained each mark. The num- ber of times each mark appears is called the frequency, and thia kind of summary is known as a frequency distribution. PICTORIAL REPRESENTATION OF DATA Most people find it easier to gain information visually from pictures rather than from the written word, although pictures 127 can bo drawn in such a way as to misrepresent the facts. Graphs, however, are most useful in statistics, and there are several kinds which may bo used : (a) circle graphs, or pie charts, (b) line graphs, (c) bar graphs, (d) rectangular distri- bution graphs, (e) dot frequency graphs, (f) histograms. 1. Pie Charts. Figure 109 shows a pie chart giving in the various sectors tho proportional amounts spent by a family from the whole income. Fig. 103 3. Histograms. The histogram looks at first sight like an upright bar chart; hut it is not used in the same circumstances, and it is constructed on a different principle. It is used when the data consists of groups of numerical measurements and when a fre- quency can be associated with any numerical range of the measurements covered by tho data. In a bar chart the length of the bars represents the frequencies, while in a histogram the area of tho columns represents the frequencies. The vertical scale of the histogram is so constructed that the areas (repre- senting frequencies) can be calculated. The vertical scale repre- sents the given frequencies divided by the numerical range of the corresponding data. This is illustrated by the following example: Example. The following table gives the salary ranges m dollars to be found in a certain factory, and the number of men whose earnings fall into each group. Plot the figures on a histogram as shown in Fig. 111. 2001-2500 15 2501-3000 5 3001-3500 3 2. Bar Graplw. Figure 110 shows the same information given in the form of a bar graph. The rectangles have the same width, but the heights of the rectangles are proportional to the amounts spent. a o o E, i- 1 t- z X lit B to 3 © I 5 Lil X 5 1 <El/> X -1 as B z w £t5 Fiu, 110 128 25 20 - 15 - IE a io * 8 $ 1 500 $ 2 OOO % 2 500 Fio. Ill S3 000 S3 500 4. Frequency Polygon. Suppose a man goes fishing and during a day catches 14 fish. Suppose tho weights of the fish to be, in lbj 4, 3£, 2, 3£, 4$, 5, 3£, 3, 2£, 4J, 4, 3, 3£, 3, These weights 129 may be tabulated as follows, according to the frequency of occurrence : lb Frequency 2 ] 21 1 3 3 Si 4 4 2 2 5 1 The results may be represented on a frequency polygon as shown in Fig. 112. The actual points plotted are joined by Straight linos. 2 - i - Fig. 112 WEIGHT, IB 5. Normal Distribution Curve. If the marks obtained in an ex- amination are plotted against their frequency, just as is done for the construction of a frequency distribution graph, a regular curve will bo obtained, provided that tho setting and marking of the quest ion papers and scripts are satisfactory. The curve is illustrated in Fig. 113. Fio. 113 50 MARKS, PER CENT 130 100 It is bell shaped and the peak should come at 50%, which is the mark ono would expect of the average child. Actually, if such a curve is plotted, it gives some indication of the standard of the paper and of the marking. If tho peak were to the left of 60%, it would mean that either the questions were too difficult or the marking was too severe. The distribution would then be said to bo skew. The same Bhapo of graph would be obtained if tho weights, or heights, of a number of people were plotted. MEAN VALUE The mean value of a set of numbers or quantities is usually understood to be the arithmetic mean. (1) If there are n numbers a^, a? s , x 3 , x <t . . . x u , then the arithmetic mean is x i + x t + x a + *i + r 1 (2) If tho frequencies of the observations x x , x 2 , x 3 , . . . x„ are/n/j./a, ,,,/„, then the mean value, or arithmetic mean is A«i + /t« 8 + /n«a + ■ ■ • +/.JCB _ £/,&; /l+/l + /» + •■• +/» = £/r' THE MODE The mode may bo regarded as the most popular value, or the moBt common member of a set of numbers or quantities. Sup- pose that a tobacconist stocked 5 different brands of cigarette. He keeps a record of the number of each brand sold, and the highest number of any particular brand sold is called the mode. For simplicity, consider 5 brands, A, B, C, D, E, and suppose that the number of each brand sold, is A, 18; B, 26; C, 36; D, 29; E, 21. Then the mode is 36. MEDIAN The median value is the value of tho middle element when tho elements are arranged in order of magnitude. Let an examination candidate gain the following marks in 8 subjects: 45, 47, 51, S6, 58, 60, 63, 69. The marks havo been arranged in order of magnitude, and the median is between 66 and 58, which is 57. Should it happen that there were an odd number of marks, add one and then divide by two in order to find the position of the middle rating. 131 RELATION BETWEEN MEAN, MEDIAN AND MODE For a normal distribution the mean, median, and mode coin, cide, but for a skew distribution they are distinct values. Fio. 114 OBSERVATIONS The positions of the values are shown in the skew distribution, Fig. 114. (i) The mean value is x lr and is the ar-coordinate of the cen- tra id of the area under the curve. (ii) The median value is a;,, and corresponds to the ordinate which bisects the area under the curve. (iii) The mode is x 3 , and is the as-coordinate corresponding to the maximum value of the frequency. The relationship between the three quantities is approxi- mately as follows : mode — median = 2 (median — mean). i.e. «, — *, = t(x s — «,). Le. a;, + 2a?, = 3ar s The arithmetic mean of several quantities may not be a typical quantity if the data include a few extreme values in one direction. The mode is sometimes thought to be the most typical score of all because it is the most frequently occurring, but it does not take into account the other values in the data. It is easy to find although there may be several other values in the set of data which satisfy the definition of a mode. The median is the middle score and it is not influenced by the other values in the data except in so far as they are either above or below the median. If a score is below the median, it is immaterial whether just below or a lot below. If the scores are concentrated in distinct and widely separated groups, the median could have little value as a measure of central tendency. 132 PERCENTILE BANK A measurement usually only has a meaning when it is com- pared with a group of similar measurements. One way of show- ing comparison b to give the rank of the score from the top. Suppose one says that Johnny is eighth in his class. This has no meaning unless it is known how many are in Johnny's class. If there were only eight pupils in the class, ho would be bottom, but if there were 250, he would have a high rank. There would be 242 below him. If a fraction is made it will bo |-JJ; changing it to a percentage, 96-8 is obtained. It is said that Johnny's percentile rank is 96-8. SCATTERING, RANGE, DEVIATION Measures ofoentral tendency such as the mean, mode, median, of a set of data give a single number which does not always give a complete picture of the data. The average rainfall of a country forms a part of the picture of the climate, but it does not tell us whether the whole rainfall occurred in three or four very heavy downpours or whether it was fairly evenly spread over several months. A more complete picture is presented if the range is stated. The rango of a set of data is the difference between the largest and smallest scores of the set. This gives a simple measure of dispersion. However, the range also has shortcomings as a measure of dispersion, and so a measure of dispersion which is related to the mean ia developed as follows : First, list each score of a sot of data, and let the score bo de- noted by X. Let the calculated mean be M . Now find out how each score differs from the mean, i.e. X — M. Square this and obtain (X — M) a . The squaring takes care of the possibility of the occurrence of negative values of X — M. Now take the arithmetic mean of the squares of the deviations by dividing the sum, S(X — M) % , of the squares by the number of scores, A 1 . This gives . Now find the square root of the result, wi _ jfvi — — . ThiB will represent a measure of the deviation of the scores from the mean. It is called the standard deviation, denoted by <?. J Therefore, f Z(X - M)* 1 V N 133 19 1. Find the mean and median of: 7, 6, 6, 5, 7, 9, 9, 8, G, 5, 4, 8, 7, 6, 3, 4, 5, 5, 4, 3. 2. Find the mean and median of: 7-7, 6-5, 4-6, 5-7, 7-4, 9-3, 6*2, 8-5, 10-7. 3. Find the moan, median, and modal number of children per family from the following table: Families IS Children per family 30 1 25 2 19 3 4. Find the mean, median, and modal number of persona per house from the following table: Hottses 20 113 120 95 60 42 21 14 5 t« Persons per house _!_ 2 3 4 5 6 7 8 9 10 ANSWERS TO EXERCISE 19 1. 6-865 6 2. 7>4; 7-4 4. 3-78; 3; 3 3. 1-85: 2; 1 134 CHAPTER TWENTY -ONE PROBABILITY The word probability sounds somewhat vague, but the sub- ject is as exact as any other branch of mathematics. It has nothing to do with such uncertainties aa predicting the weather, but deals with such probloms as the chance o f, say, picking an ace from a pack of cards. There is a great deal of betting and gambling nowadays: peoplo'a attitude is that 'someone must win'. But the mathematical ehancea of winning, perhaps a foot- ball pool, are negligible. Mathematicians have decided upon the following scale for measuring probability: I means that the occurrence is bound to happen; means that the occurrence cannot happen. Probabilities are written in decimal or frac- tional form. A probability of one in ten, therefore, could be written as 0-1 or £f, So trie probability of tossing a coin to come down heads, or tails, could be 0-5 or $; the probability of pick- ing diamonds from a pack of cards is 0'25 or J. Example 1. Find the probability of getting (i) two heads, (ii) a head and a tail, with two throws of a coin. There are the following possibilities: First throw Second throw H H B T T B T T (i) Thus there is 1 possibility out of 4 that two heads are thrown. So the probability is J or 0'25. (ii) There are 2 possibilities in 4 that a head and a tail are thrown. So the probability is J or \ or 0-5. Example 2. Find tho probability of throwing (i) three heads, (ii) two heads and one tail, (iii) three heads or three tails, with the tossing of three coins. There are the following possibilities : First throw Second throw Third throw H B B H T H H B T H T T T H H T T B T B T p Jt w ITM — K 135 It will bo seen t lint (i) the probability of throwing three heads is J, (ii) the probability of throwing two heads and a tail is f, (iii) the probability of throwing three faces alike is i + i = i = i- THE ADDITION LAW This law states that if the probabilities of n naturally ex- clusive events happening are p lr p s , p 3 , p A , . . . p„, then tho probability that one of the events will occur is P = Pi + Pa + Pi + P« + ■ • • + Pn- Example I. Find tho probability of throwing a 5 or a 6 with an ordinary die (singular of dice). The probability of tossing n ~> is I . The probability of tossing a ii is j. Therefore the probability of tossing a 5 or a is J + \ = J. Example 2. Find the probability of tossing not more than 4 with ono toss of a die. Tho probability of throwing 1 is J Tho probability of throwing 2 is f The probability of throwing 3 is -j The probability of throwing 4 is £. Therefore the probability of throwing either 1 or 2 or S or 4 is * + * + i + J = f = f . Example 3. Let a box contain Q black balls and 4 white balls. Find tho probability of taking out a black or a whito ball. Tho probability of taking out a black ball is -?g. The probability of taking out a whito ball is -fa. Therefore the probability of taking out a black or a white ball is i'o "I" iV = Jo = !• This is the expected result, because the box contains only white and black balls. THE MDXTIPLICATION LAW This law states that if the probabilities of n independent events are p l? p 2 , p 3 , p A , . . . p n , then the probability of all the events occurring is P = Pi X p a X p s X p t X . . . X p„. Events are independent when no event can influence any future events. 136 Example 1. Find the probability that 4 heads will be thrown in ~> throws of a coin. The probability of throwing a head on tho 1st throw = -J- Tho probability of throwing a head on tho 2nd throw = 4 Tho probability of throwing a head on tho 3rd throw = A Tho probability of throwing a hend on the 4th throw = J Tho probability of throwing a head on the 5th throw = k. Therefore the possibility of throwing five heads in a row is J x i x J x J x i = -ii. Example 2. Find tho probability of throwing two 3's with two throws of a dice. The probability of throwing 3 on the 1st throw = £ The probability of throwing 3 on the 2nd throw = f. Therefore, the probability of throwing two 3's is J x i = -gg. Example 3. A die is thrown twice. Find tho probability that tho first throw is less than, or equal to, 4 and the second throw is less than, or equal to, 3. The probability of the first throw being less than, or equal to, 4 is *, The probability of the second throw being less than, or equal to, 3 is f . Therefore, tho probability of both ovents happening is 4 3 12 _ 1 30 ~~ 3" 6*8 Till-; PROBABILITY TREE This is a method of calculating probabilities when Micro is an equal chance of a sample being rejected or accepted. Tho probabilities of tho event's happening are indicated on tho various branches; tho final probability of its happening is ob- tained by multiplying the final probability in the branch. The method can be used only when one stnglo event can occur at a time, that is, when the events are mutually exclu- sive. Consider the tossing of a coin, H means that a head ap- pears, and T means that a tail appears. Figure 116 shows the tree obtained by the tossing of a coin. The tree shows that the probability of (i) three tails occurring iB£x-Jxl = £ (ii) a tail and 2 heads occurring is (J x i x i) + (J X i x t) + (i x i x 4) - J. 137 FlQ. 115 EXEBOISE 20 1 . Two dollars and four other coins are put in line at random. Find the probability that the end coins are both dollars. 2. Nine balls are drawn at random from a bag containing 1 1 white and 9 black balls. Find the probability that 5 are white. 3. Ten cents and two dollars are placed in a circle at random. Find the odds against the two dollars being placed together. 4. Find the probability of getting a double six at least once in ten throws of two dice. 5. A bag contains 8 white balls and 6 black balls. Five balls are drawn at random. Find the probability that (i) three balls are white, (ii) three or more balls are white. ANSWERS TO EXERCISE 20 i - is 4. 0-21,-1 58,212 2. „' =2z 0-347 167,960 3. 0:2 °- (l) 2002* (ii) Kilfi 2002' LIST OF SYMBOLS { } the set of e is a member of; belongs to £ does nob belong to 3 for soi no numbers £ A set A n B the intersection of A and B A' the complement of A A-* B A'm mapped on to B V A 1 p and 5 p-*- q p implies q OX vector OX A transposed matrix of A a standard deviation z> implies Z+ tho set of positive integers y for any numbers a' the inverse of a I identity element the null set c is contained in A \J B tho union of A and J3 U tho universal set (universe of discourse) ~ p negation of p P V <? p or g or both p t-y q p implies q and q implies p A -1 inverse of A a vector a =3 has as one of its subsets ■o- implies and is implied by © operation. 138 139 DEFINITIONS Abellan group algebra associative property binary operation binary scale Boolean algebra closure property commutative property conjunction de Morgan's laws disjunction distributive property domain of a function function field A group which has a binary opera- tion which is commutative. A system is an algebra if ib is pro- vided with binary operations of addition and multiplication, and a scalar multiplication which make it both a vector space and a ring. For (i) addition J* + ff) + s = x + {y + z) (ii) multiplication (at x y) x e = ar x (y x a). An operation combining two ele- ments. A number syBtem containing only and 1. A class of statements and their logioal relations. The algebra of sets. The property that the result of com- bining any two elements is also a member of the set. For (i) addition x + y — y + x. (ii) multiplication x * V = V x *• Both a and 6. (i) (o + 6)' = a'b' (ii) (ai)' = a'b'. Eilhor a or b or both. Multiplication is distributive with respect to addition if xx(y+z)=xxy + xxz. The set of all the first members of the ordered pairs in the function. A relation in which no two different pairs have the samo first member, A ring becomes a field if it baa a unity element for multiplication and contains a reciprocal for every element except 0. no group identity property inverse operation isomorphic group law of absorption mapping many- to -one corre- spondence matrix singular matrix transpose of a matrix A system is a group if it has a binary operation that is associa- tive, has an identity element for the operation, and has an inverse for every element. That operation which leaves an cle- ment as before. I is an identity element for the operation @ if I © X = X © I = X, for every X. J is for addition and 1 for multiplication. A is the inverse of B with respect to the operation © whose identity element is I, if A®B = B@A = I. For (i) addition, a + (— a) — 0, where (— a) is the additive inverse of a. The inverse of an integer is its negative, (ii) multiplication, sx- = l,a#0, where - a a is the multiplicative inverse of a. The inverse of an in- teger is its reciprocal. Two groups are isomorphic if they have the same structure or form. a + ab = a. A function. A single object is the image of more than one object. An array of elements. An operator. A matrix whose determinant is 61 The matrix is sin- gular if ad — bo = 0. It A = b d , the transposed matrix 141 unit matrix zero matrix modulo arithmetic negative one-to-one corre- spondence ordered pair range of a function reciprocal relation ring set complement of set A elements of a set intersection of two sets A and B null set subset union of two sets A and B universal set [o oj" A finite urithitK'tiu. Kur uxamplc, the integers modulo 4 arc 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, . . . A is the negative of B if A +B = B +A=0. Each element of the range is matched with one and only one element of the domain. A pair of elements arranged in a certain order. The eot of all tho second members of the ordered pairs in the function. A is the reciprocal of B if A.B =B.A = 1. A set of ordered pairs. A system is a ring if ib has two associated binary operations called addition and multiplication, is an Abolian group with respect to ad- dition, and if multiplication is dis- tributive with respect to addil i> at, A collection of elements distin- guishable from non members and from each other. Those elements in the universal sot which are not in sot A. The members of a set. Those elements which are in both set .4 and set B. Tho empty sot, 0, or { }. A is a subset of set B if every ele- ment of A is also an clement of sot B. Another set of elements which are in set A or set B or both. Thcsotofallthcolements, called U. 142 truth table unity element vector vector space Venn diagram zero clement A table showing whether or not a sentence is logically true. U is a unity element for multipli- cation if UxX=XxU = X, for every X. A directed line segment. A system is a vector space if it is an Abelian group with respect to addition, is subject to a scalar mul- tiplication by elements from an associated field of scalers, and if the scalar multiplication obeys the laws: r.(a + b) = r.a + r.b (r + a). a = r.a + B.a r.(s.a) = (r.s).o l.a = a. A closed curve used to represent a sot. A is a zero elemont for addition if A -I- X - X + A = X, for every X. 143 14 $*{<>#> Titles available in the NEWNES INTRODUCTORY SERIES are; — INTRODUCTION TO ALGEBRA 1 2s. 6d. cut flush, IBs. cased. INTRODUCTION TO ANATOMY 12s. 6d. cut flush, 18s. cased. INTRODUCTION TO BOTANY 12s. 6d. cut flush, IBs. cased. INTRODUCTION TO CALCULUS 15s. cut flush, 21 s. cased. INTRODUCTION TO CHEMISTRY (2s. 6d. cut flush, 18s. cased. INTRODUCTION TO MATHEMATICS 12s. 6d, cut flush, 18s. cased. INTRODUCTION TO MECHANICS /2s. 6d. cut flush, 18s. cased. INTRODUCTION TO PHYSICS /2s. 6d. cut (tush, 18s. cased. INTRODUCTION TO PSYCHOLOGY 10$. 6d. cut flush, /5s, cased. INTRODUCTION TO RADAR AND RADAR TECHNIQUES IDs. 6d. cut flush, 15s, cased. INTRODUCTION TO RADIO ASTRONOMY 12s. 6d. cut flush, 18s, cased. FROM ALL BOOKSELLERS A fully descriptive leaflet is available on re- quest from the publishers, GEORGE NEWNES LTD., Tower House, Southampton Street, London, W.C.2. EDUCATION C. C. T. Baker wu bom in Monmouthshire in November 1907. lie was educated at Aberdare Grammar School, 1921-1926, and at The University College of Wales, Aberystwyth, 1926-1930, and was a scholarship holder at both places. He ohtained the degree of B.Sc. in Mathematics and Physics in 1929, and the Diploma' in Education in 1930. Since 1130, the author has had a wide, varied and extensive experi- ence teaching mathematics and physics, at all levels, in Grammar Schools and Technical Colleges. He is also the author of a number of other popular and successful brinks on mathematics, which are widely used in this country and abroad.